Some days ago it came to my attention the fact that there is a religious group that considers the presence of propositions within the Old Testament that seem to describe events that have happened after the propositions were written as proof of the divine origin of such a text. Not only that: to my understanding, they seem to infer from those successful prophecies that also the other predictions present in the Bible must turn out to happen somewhere in the future.
Now, in these statements, there are several problems. One can note, for instance, that successful scientific theories like the Newtonian gravitation continuously predict with great accuracy the future, and yet this doesn’t seem to be generally considered proof that Isaac Newton had a divine nature. On the other hand, the fact that his gravitational law predicts the orbits of the Earth, of Mars, of the modules of the Apollo missions, etc., doesn’t prevent the same theory to fail when it comes to the orbit of Mercury. So, this single counter-example might be considered enough to prove that the first paragraph of this article contains false statements.
But then there is another problem. How do I evaluate whether a prophecy is true or false? Despite the obvious difficulties due to the fact that this kind of predictions are expressed in a human language (translated from one tongue into another) and not in mathematical language and as such ambiguous by its very nature, one can still try – I reasoned – to apply the standard method used in science. In other words, one can calculate the probability of the null hypothesis (that, in this case, is: the prophecy is true by chance) and see what kind of number he gets.
Well, I discovered right away that this calculation had in fact been attempted by Peter Stoner, a math teacher at Pasadena City College (R), and it can be found in his book, Science Speaks (freely available here). Let’s consider one of those prophecies, the one that seems to be particularly close to the heart of Marina, the Jehovah’s Witness who is currently persecuting me, the prophecy of Babylon:
And Babylon … shall never be inhabited, neither shall it be dwelt in from generation to generation: neither shall the Arabian pitch tent there; neither shall the shepherds make their fold there. But wild beasts of the desert shall lie there; and their houses shall be full of doleful creatures.
(Isa. 13:19-21 – written 712 B.C.)
And they shall not take of thee a stone for a corner, nor a stone for foundations; but thou shalt be desolate forever, saith the Lord … Neither doth any son of man pass thereby.
(Jer. 51:26,43 – written 600 B.C.)
Now, in his book, Peter Stoner translates this prophecy in the following seven events or propositions:
: Babylon shall be destroyed. : It shall never be reinhabited (once destroyed). : The Arabs shall not pitch their tents there (once destroyed). : There shall be no sheepfolds there (once destroyed). : Wild beasts shall occupy the ruins (once destroyed). : The stones shall not be taken away for other buildings (once destroyed). : Men shall not pass by the ruins (once destroyed).
I have added the expression within brackets, to enhance clarity. The probabilities of the single events (on the right) are those proposed by Peter Stoner, and I am not going to discuss them. Now, if we indicate the whole prophecy with E, according to Stoner the probability of the null hypothesis is
This is a really small probability for the null hypothesis, so we must accept that a power beyond human comprehension has… But let’s read again the prophecy: it seems articulated but, in fact, it only says that the place where Babylon once raised will be basically abandoned by human beings. In other words, it seems reasonable to say that for . If that is the case, then we have
A further observation is that if the place is desolated with no human beings () then it is reasonable to assume that it becomes the reign of wild animals. In other words: . Not only that, I would guess that it is safe to assume that is about one. Then we have found
In other words, the mistake that leads Stoner to a completely wrong value for is the fact that he considered the events as independent one from the other, while this is obviously not the case.
Now, is this the probability of the null hypothesis? Well, it depends, because this is a case in which we have a prediction that has been got from a very long book with thousands of propositions, some of which look very much like predictions. Now, of course, when one picks a prophecy among all these propositions, he might be unconsciously tempted to pick the one that looks more like a fulfilled prophecy. In other words, we have to check for multiple comparisons in this case. So, let us consider that we have a number of N propositions similar to the one about Babylon. The probability that at least one of these propositions is true purely by chance is
The function is plotted by the following code and as you can see, for N >4 we have that the null hypothesis becomes very likely. In other words, if we pick this prophecy among 5 other similar sentences, its resemblance to reality is just due to chance. In the figure below the red line indicates a probability of 0.05. A probability equal to or higher than 0.05 is associated with null hypotheses that must be considered true.
It should be added that the fact that the prophecy of Babylon has to be considered true is highly questionable. The reader can do his own research on that, for example starting from the notes collected in this text.
% file name = Babylon % date of creation = 20/03/2021 % it calculates the probability that N prophecies are true by chance clear all % array for p(N) N = 30; for i=1:N p(i) = 1-(99/100)^i; p2(i) = 0.05; endfor % plotting plot ([1:N],p(:),’-k’, “linewidth”, 2) hold on plot ([1:N],p2(:),’-r’, “linewidth”, 2) xlabel(‘number of prophecies’); ylabel(‘probability of the null hypothesis’); grid minor grid on
When Ettore Majorana first met Enrico Fermi, between the end of 1927 and the beginning of 1928, Fermi – who was already an acclaimed scientist in the field of nuclear physics – had just solved an ordinary differential equation of the second-order (whose solution is now commonly named the Thomas-Fermi function) – by numerical integration. It required a week of assiduous work for him to accomplish this task, with the aid of a hand calculator. Fermi showed the results (a table with several numbers) to Majorana, who was a 21 years old student of electrical engineering who had some vague idea of switching from the field of boring ways of providing electrical energy for boring human activities, to the quest for the intimate structure of matter, under the guide of Fermi, the brightest Italian scientific star of that period.
Majorana looked at the numerical table, as I wrote, and said nothing. After two days he came back to Fermi’s lab and compared his own results with the table made by Fermi: he concluded that Fermi didn’t make any mistake, and he decided that it could be worth working with him, so he switched from engineering to physics (Segrè E. 1995, page 69-70).
Only recently it has been possible to clarify what kind of approach to the equation Majorana had in those hours. It is worth mentioning that he not only solved the equation numerically, I guess in the same way Fermi did but without a hand calculator and in less than half the time; he also solved the equation in a semianalytic way, with a method that has the potential to be generalized to a whole family of differential equations and that has been published only 75 years later (Esposito S. 2002). This mathematical discovery has been possible only because the notes that Majorana wrote in those two days have been found and studied by Salvatore Esposito, with the help of other physicists.
I won’t mention here the merits that Majorana has in theoretical physics, mainly because I am very very far from understanding even a bit of his work. But as Erasmo Recami wrote in his biography of Majorana (R), a paper published by Majorana in 1932 about the relativistic theory of particles with arbitrary spin (Majorana E. 1932) contained a mathematical discovery that has been made independently in a series of papers by Russian mathematicians only in the years between 1948 and 1958, while the application to physics of that method – described by Majorana in 1932 – has been recognized only years later. The fame of Majorana has been constantly growing for the last decades.
The notes that Majorana took between 1927 and 1932 (in his early twenties) have been studied and published only in 2002 (Esposito S. et al. 2003). These are the notes in which the solution of the above-mentioned differential equation has been discovered, by the way. In these 500 pages, there are several brilliant calculations that span from electrical engineering to statistics, from advanced mathematical methods for physics to, of course, theoretical physics. In what follows I will go through what is probably the least difficult and important page among them, the one where Majorana presents an approximated expression for the maximum value of the largest of the components of a normal random vector. I have already written in this blog some notes about the multivariate normal distribution (R). But how can we find the maximum component of such a vector and how does it behave? Let’s assume that each component has a mean of zero and a standard deviation of one. Then we easily find that the analytical expressions of the cumulative distribution function and of the density of the largest component (let’s say Z) of an m-dimensional random vector are
We can’t have an analytical expression for the integral, but it is relatively easy to use Simpson’s method (see the code at the end of this paragraph) to integrate these expressions and to plot their surfaces (figure 1).
Now, what about the maximum reached by the density of the largest among the m components? It is easy, again, using our code, to plot both the maximum and the time in which the maximum is reached, in function of m (figure 2, dotted lines). I have spent probably half an hour in writing the code that gives these results, but we usually forget how fortunate we are in having powerful computers on our desks. We forget that there was a time in which having an analytical solution was almost the only way to get mathematical work done. Now we will see how Majorana obtained the two functions in figure 2 (continuous line), in just a few passages (a few in his notes, much more in mine).
% file name = massimo_vettore_normale
% date of creation = 22/05/2019
delta = 0.01;
n(1) = 0.
n(i) = delta + n(i-1);
n_2(i) = - n(i);
f(i) = 0.39894228*( e^( (-0.5)*( n(i)^2 ) ) );
sigma(1) = 0.;
sigma(3) = sigma(1) + delta*( f(1) + ( 4*f(2) ) + f(3) )/3;
sigma(2) = sigma(3)*0.5;
sigma(j+2) = sigma(j) + delta*( f(j) + ( 4*f(j+1) ) + f(j+2) )/3;
F(i) = 0.5 + sigma(i);
F_2(i) = 1-F(i);
m(i) = i;
F_Z (i,j) = F(i)^j;
F_Z_2 (i,j) = F_2(i)^j;
f_Z (i,j) = 0.39894228*j*( F(i)^(j-1) )*( e^( (-0.5)*( n(i)^2 ) ) );
f_Z_2 (i,j) = 0.39894228*j*( F_2(i)^(j-1) )*( e^( (-0.5)*( n(i)^2 ) ) );
I have always been fascinated by integrals since I encountered them a lifetime ago. I can still remember the first time I learned the rule of integration by parts. I was caring for my mother who was dying. That night I was in the hospital with her, but she couldn’t feel my presence, she had a tumour in her brain and she was deteriorating. And yet I was not alone, because I had my book of mathematics and several problems to solve. But when my mind was hit by the disease for the first time, about a year later, and I lost the ability to solve problems, then real loneliness knocked at my door.
Now, why am I talking about the integration by parts? Well, I have discovered a few days ago, while studying Majorana’s notes, that integration by parts – well known by students to be a path towards recursive integrations that usually leads to nowhere – is, in fact, a method that can be useful for developing series that approximate a function for large values of x (remember that Taylor’s polynomials can approximate a function only for values of x that are close to a finite value , so we can’t use them when x goes to ∞). Majorana used one such a series for the error function. He developed a general method, which I tried to understand for some time, without being able to actually get what he was talking about. His reasoning remained in the back of my mind for days, while I moved from Rome to Turin, where I delivered a speech about a paper on the measure of electric impedance in the blood of ME/CFS patients; and when I cried, some minutes later, looking at my drawings put on the screen of a cinema, Majorana was with me, with his silence trapped behind dark eyes. A couple of days later, I moved to a conference in London, searching for a cure that could perhaps allow my brain to be normal again and I talked with a famous scientist who worked on the human genome project. Majorana was there too, in that beautiful room (just a few metres from Parliament Square), sitting next to me. I could feel his disappointment, I knew that he would have found a cure, had he had the chance to examine that problem. Because as Fermi once said to Bruno Pontecorvo, “If a problem has been proposed, no one in the world can resolve it better than Majorana” (Esposito S. et al. 2003). Back in Rome, I gave up with the general method by Majorana and I found the way to calculate the series from another book. The first tip is to write the error function as follows:
Now by integrating by parts, one gets
But we can integrate by parts one other time, and we get
And we can go on and on with integration by parts. This algorithm leads to the series
whose main property is that the last addend is always smaller (in absolute value) than the previous one. And even though this series does not converge (it can be easily seen considering that the absolute value of its generic addend does not go to zero for k that goes to ∞, so the Cauchy’s criteria for convergence is not satisfied) it gives a good approximation for the error function. From this series, it is easy to calculate a series for the Gaussian function (which is what we are interested in):
A clever way to solve a transcendental equation if you don’t want to disturb Newton
Taking only the first two terms of the series, we have for the cumulative distribution function of Z the expression:
The further approximation on the right is interesting, I think that it comes from a well-known limit:
Now we can easily calculate the density of Z by deriving the cumulative distribution function:
With a further obvious approximation, we get:
In order to find the value of x in which this density reaches its largest value, we have to search for the value of x in which its derivative is zero. So we have to solve the following equation:
Which means that we have to solve the transcendental equation:
Majorana truncated the second member of the equation on the right and proposed as a solution the following one:
Then he substituted again this solution in the equation, in order to find ε:
With some further approximations, we have
So Majorana’s expression for the value of x in which the density of Z reaches its maximum value is
I have tried to solve the transcendental equation with Newton’s method (see the code below) and I found that Majorana’s solution is a very good one (as you can see from figure 3). Now, If we compare the approximation by Majorana with what I obtained using numerical integration at the beginning (figure 2) we see that Majorana found a very good solution, particularly for the value of . Note: the transcendental equation that has been solved here seems the one whose solution is the Lambert W function, but it is not the same!
% file name = tangenti
% date of creation = 08/06/2019
x(1) = 1; %the initial guess
m(i) = i;
for j = 2:1:1000
f(j-1) = exp( 0.5*( x(j-1)^2 ) ) - ( m(i)/( x(j-1)*sqrt(2*pi) ) );
f_p(j-1) = x(j-1)*exp( 0.5*( x(j-1)^2 ) ) + ( m(i)/( (x(j-1)^2)*sqrt(2*pi) ) );
x(j) = x(j-1) - ( f(j-1)/f_p(j-1) );
if ( abs(x(j)) < 0.001 )
max_t (i) = x(j);
% the aproximations by Majorana
max_t_M (j) = sqrt(log(j^2)) - ( log(sqrt(2*pi*log(j^2)))/sqrt(log(j^2)) );
% it plots the diagrams
plot(m(1:1:100),max_t (1:1:100),'.k','Linewidth', 1)
ylabel('time for maximum value')
plot(m(1:1:100),max_t_M (1:1:100),'-k','Linewidth', 1)
legend('numerical integration',"Majorana's approximation", "location", 'southeast')
From 1934 to 1938 Majorana continued his studies in a variety of different fields (from game theory to biology, from economy to quantistic electrodynamics), but he never published again (R), with the exception for a work on the symmetric theory of electrons and anti-electrons (Majorana E. 1937). But it has been concluded by biographers that the discoveries behind that work were made by Majorana about five years earlier and yet never shared with the scientific community until 1937 (Esposito S. et al. 2003). And in a spring day of the year 1938, while Mussolini was trying his best to impress the world with his facial expressions, Ettore became a subatomic particle: his coordinates in space and their derivatives with respect to time became indeterminate. Whether he had lived in a monastery in the south of Italy or he had helped the state of Uruguay in building its first nuclear reactor; whether he had seen the boundless landscapes of Argentina or the frozen depth of the abyss, I hope that he found, at last, what he was so desperately searching for.
He had given his contribution to humanity, so whatever his choice has been, his soul was already safe. And as I try to save my own soul, going back and forth from mathematics to biology, in order to find a cure, I can feel his presence. The eloquence of his silence trapped behind dark eyes can always be clearly heard if we put aside the noise of the outside world. And it tells us that Nature has a beautiful but elusive mathematical structure which can nevertheless be understood if we try very hard.
In the meanwhile, I write these short stories, like a mathematical proof of my own existence, in case I don’t have further chances to use my brain.
In 1999 I was wandering in Pisa with a booklet in the pocket of a worn coat, too short for my frame. That coat was dark blue on the outside, green and red inside, with one mended sleeve and an austere cowl: I was much like a young monk, with his holy book (figure 1). I can remember neither its title nor the author, though. It was an introduction to statistical thermodynamics, with beautiful figures, a coloured cover, and less than 100 pages. It contained the work by Maxwell on the kinetic theory of ideal gasses, along with other material. I borrowed it from the University Library because I was fascinated by the way in which Maxwell was able to describe the properties of gasses with just a few hypotheses and some relatively easy mathematical passages. I felt that there was an enormous attraction in these methods, I realized with pleasure that math could give the power to completely understand and hold in hand physical systems and even, I started speculating, biological ones.
My second favourite composer back then was Gustav Mahler (the favourite one being Basil Poledouris): he represented my own way to classical music and I chose him because he wasn’t among the musicians my father and my brother shared a love for. I felt, during my teens, that I had to find my private space, and I met it one day on a used book stand: a cassette tape of Das Lied von The Erde, with a few sheets containing the translation to Italian of the songs. Mahler was born in 1860, a few weeks after Maxwell published his pivotal work about ideal gasses in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (R) (figure 2). But in 1999 I was repeatedly listening to a CD with a collection of songs sung by Edith Piaf and Charles Trenet, because I was studying French, and I was having a hard time with pronunciation. So, imagine a secular monk in his prime who listens to old French songs while keeping one hand on a book of statistical thermodynamics hidden in his pocket, wherever he goes, wandering in the streets of Pisa, a city which gave birth to Galileo Galilei. This seems a beautiful story, much like a dream, right? Wrong.
I had already started my struggle against the mysterious disease that would have completely erased my life in the following years. In the beginning, it had a relapsing-remitting course, so I could sometimes host the hope that I was recovering, only to find myself caught by the evil curse again. At the end of the year 1999, I was losing my mind, I knew that and I was also aware that my holy book couldn’t save me. I clearly remember one evening, I was walking on Ponte di Mezzo, a simple and elegant bridge above the Arno, and I felt that I couldn’t feel sorrow for the loss of my mind: I realized that not only the functions of my brain assigned to rational thinking were gone, but my feelings couldn’t function properly either. In fact, I noted without a true manifestation of desperation that I had lost my emotions. One day, after spending in vain about eleven hours on a single page of a textbook of invertebrate palaeontology, I accepted that I couldn’t read anymore, at least for the moment.
Had I known for sure that I wouldn’t have recovered in the following twenty years, I would have quite certainly taken my own life, jumping from a building; a fate that I have been thinking about almost every day ever since. I considered this possibility during the endless sequence of days in which there has been nothing other than the absence of my thoughts.
The distribution of velocities of an ideal gas and the one hundred years gap
In the already mentioned paper by Maxwell, he derived the probability density of the speed of the molecules of a gas, granted that the three components of the vector of speed are independent random variables (hyp. 1) and that they share the same density (hyp. 2), let’s say f. Moreover, the density of the speed has to be a function only of its module (hyp. 3). These three hypotheses together say that there is a function Φ such that
This is a functional equation (i.e. an equation in which the unknown is a function) whose solution is not detailed in Maxwell’s work. But it can be easily solved moving to polar coordinates (see figure 3) and deriving with respect to θ both members (the second one gives naught since it depends only to the distance from the origin).
Another way to solve the functional equation is to use the method of Lagrange’s multipliers, searching for the extremes of the density of the velocity, when its module is fixed. In either case, we obtain the differential equation:
which leads to the density for each component of the speed:
where σ can’t be determined using only the three hypotheses mentioned above. Considering then the well-known law of ideal gasses (pV=nRT) and an expression for p derived from the hypothesis that the collisions between the molecules of gas and the container are completely elastic, Maxwell was able to conclude that:
where m is the mass of the molecule of gas, T is the absolute temperature and K_B is the Boltzmann’s constant. It was 1860, Mahler’s mother was going to deliver in Kaliště, Charles Darwin had just released his masterpiece “On the origin of species”, forced to publish much earlier than what he had planned because of the letter he had received from Wallace, in which he described about the same theory Darwin had been working on for the previous 20 years. In the same point in time, Italy was completing its bloody process of unification, with the Mille expedition, led by Giuseppe Garibaldi.
But the functional equation I have mentioned at the beginning of this paragraph has brought with it a mystery for years, until 1976, when an employee at General Motors Corporation published a short note in the American Journal of Physics [R] in which he showed how Maxwell’s functional equation is, in fact, an example of the well known Cauchy’s functional equation:
In order to prove that, you just have to consider the following definition:
The name of the mathematician who made this observation is David H. Nash, and he has the merit of finding something new in one of the most known equation of physics, an equation that is mentioned in every book of thermodynamics, an equation that has been considered by millions of students in more than a century. It was 1976, my mother was pregnant with my brother; Alma, Gustav Mahler’s wife, had died about ten years before.
Module of random vectors
Once Maxwell found the density of probability for each component of the speed of the molecules of an ideal gas, he searched for the density of the module of the speed. There is a relatively simple way of doing that. With the following notation
we have that the repartition function of Z is given by the integrals of the density of X within the sphere in figure 4. We have:
The second expression is the same as above but in polar coordinates. Then we can obtain the density of Z by derivation of the repartition function. And this method can be extended to an m-dimensional space. This was the method used by Maxwell in his paper. And yet, there is another way to obtain the expression of the module of a random vector: I have explored it in the last months, during the rare hours in which I could function. By the way, only in the Summer of 2007 I was able to study the kinetic theory by Maxwell, eight years after I borrowed the holy book. Such a waste.
The hard way towards the density of the module of a random vector
When a student starts studying statistics, she encounters a list of densities: the normal distribution, the gamma distribution, the exponential distribution etc. Then there are several derived distributions that arise when you operate sums, roots extractions etc. on random variables. In particular, if f_X is the densty of X and Y = X², then we have
On the other hand, if Y = √X we have
Another important result that we have to consider is that given
By using these results I have been able to find that the expression of the density for the module of an m-dimensional random vector is:
In particular, for m = 3 we have
The caseof normal random vectors: the modified Bessel function
In particular, if the random vector has dimension 3 and its components are normal random variables with the same expected value and variance, we have that the density of its module is given by
where I_0 is the modified Bessel function, which is one solution of the differential equation:
whose name is modified Bessel equation. The integral expression of the modified Bessel function is:
I have coded a script in Matlab which integrates numerically this function (available here for download) which plots the surface in figure 5 and also gives the following table of values for this function.
The following is the flowchart of the script I coded.
The caseof normal random vectors with a naught expected value: the upper incomplete gamma function
If we consider random variables with an average that is zero (this is the case with the components of speed in ideal gasses), then the density is given by
which is a Chi distribution with 3 degrees of freedom, scaled with a scale parameter given by s = 1/σ. In the expression of the repartition function, ϒ is the lower incomplete gamma function, which is defined as follows:
I have written a code for its numerical integration (available here for download), the output of which is in figure 6.
Conclusion, twenty years later
The density of the module of the velocity of the molecules of an ideal gas is, in fact, a scaled Chi distribution with 3 degrees of freedom, and it is given by
It can be numerically integrated with the following script I made for Octave/Matlab, which gives the plot in figure 7. Another similar script gives the plot in figure 8. These plots represent the Maxwell-Boltzmann distribution, the centre of the holy book that an unfortunate boy was carrying in his pocket, all alone, some twenty years ago. He could have easily died by his own hand in one of the several thousand days of mental and physical disability that he had to face alone. Instead, he has survived. Had it been only for finding the Maxwell-Boltzmann distribution following another path, it would have been worth it. But he has found much more, including a bright girl, the magnificent next stage of evolution of human beings.
% file name = legge_Maxwell-Boltzmann_2
% date of creation = 22/02/2019
% it plots the density and the distribution function for the
% Maxwell-Boltzmann distribution considered as a function of temperature
% and speed
% we define some parameters
K_B = 1.381*10^(-23) % Boltzmann's constant
m = 4.65*10^(-26) % mass of the molecule N_2
% we define the array of temperature from 0° C to 1250° C
T (1) = 273.15
for i = 2:1:1250
T (i) = T (i-1) + 1.;
% it defines f_gamma in 3/2
f_gamma = sqrt(pi)/2.
% delta of integration
delta = 1.0
% it defines the array for the abscissa
z (1) = 0.;
for i = 2:1:2500
z (i) = z(i-1)+delta;
% it defines the density
for j = 1:1:1250
% it defines a constant
c = ( m/(K_B*T(j)) );
for i = 1:1:2500
f (j,i) = ( c^1.5 )*sqrt(2./pi)*( z(i)^2. )*( e^( -0.5*c*z(i)^2. ) );
% it calculates the ripartition function for He
F (j,1) = 0.;
F (j,3) = F (j,1) + delta*( f(j,1) + ( 4*f(j,2) ) + f(j,3) )/3;
F (j,2) = F (j,3)*0.5;
F (j,k+2) = F (j,k)+delta*( f(j,k)+( 4*f(j,k+1) )+f(j,k+2) )/3;
% It plots f and F
mesh(z(1:100:2500), T(1:100:1250), f(1:100:1250,1:100:2500));
mesh(z(1:100:2500), T(1:100:1250), F(1:100:1250,1:100:2500));
In a previous blog post, I have discussed the joint density of a random vector of two elements that are normally distributed. I was able to prove the expression for the joint probability, not without fighting against some nasty integrals. In the end, I introduced the expression of the joint probability for a random vector of m normally distributed elements and I left my four readers saying “I have no idea about how it could be proved“. We were in June, I was in the North of Italy back then, hosted by friends but mainly alone with my books in a pleasant room with pink walls and some dolls listening to my speculations without a blink; a student of engineering was sharing with me, via chat, her difficulties with the surprisingly interesting task of analyzing the data from some accelerometers put in the mouth of fat people while they stand on an oscillating platform; my birthday was approaching and I was going to come back in Rome after a short stop in Florence, where I was for the first time fully aware of how astonishingly beautiful a woman in a dress can be (and where I saw the statues that Michelangelo crafted for the Tomb of the Medicis, the monument to Lorenzo in particular, which is sculpted in our culture, more profoundly than we usually realize).
But on a day in late December, while I was planning my own cryopreservation (a thought I often indulge in when my health declines even further), I realized that the covariance matrix is a symmetrical one so it can be diagonalized, and this is the main clue in order to prove the expression of this density. As obvious as it is, I couldn’t think of that when I first encountered the multivariate normal distribution, and the reason for this fault is my continuous setbacks, the fact that for most of the last 20 years I have not only been unable to study but even to think and read. And this is also the reason why I write down these proofs in my blog: I fear that I will leave only silence after my existence, because I have not existed at all, due to my encephalopathy. I can’t do long term plans, so as soon as I finish a small project, such as this proof, I need to share it because it might be the last product of my intellect for a long time. So, what follows is mainly a proof of my own existence, more than it is a demonstration of the multivariate normal distribution.
Before introducing the math, two words about the importance of the multivariate normal distribution. Many biological parameters have a normal distribution, so the normal density is the most important continuous distribution in biology (and in medicine). But what happens when we are considering more than one parameter at the time? Suppose to have ten metabolites that follow a normal distribution each, and that you want to calculate the probability that they are all below ten respective maximal values. Well, you have to know about the multivariate normal distribution! This is the reason why I believe that anyone who is interested in biology or medicine should, at least once in her lifetime, go through the following mathematical passages.
Can’t solve a problem? Change the variables!
In this paragraph, I present a bunch of properties that we need in order to carry out our demonstration. The first one derives directly from the theorem of change of variables in multiple integrals. The second and the third ones are a set of properties of symmetrical matrices in general, and of the covariance matrix in particular. Then, I collect a set of integrals that have been introduced or calculated in the already cited blog post about the bivariate normal distribution. The last proposition is not so obvious, but I won’t demonstrate it here, and those who are interested in its proof, can contact me.
PROPOSITION 1 (change of variables). Given the continuous random vector X = () and the bijective function Y = Φ(X) (figure 1), where Y is a vector with m dimensions, then the joint density of Y can be expressed through the joint density of X:
PROPOSITION 2 (symmetrical matrix). Given the symmetrical matrix C, we can always write:
where are the eigenvalues of matrix C and the columns of P are the respective eigenvectors. It is also easy to see that for the inverse matrix of C we have:
Moreover, the quadratic form associated with the inverse matrix is
PROPOSITION 3 (covariance matrix). If C is the covariance matrix of the random vector X = (), which means that
then, with the positions made in Prop. 2, we have
where is the standard deviation of and is the correlation coefficient between and .
PROPOSITION 4 (some integrals). It is possible to calculate the integrals in the following table. Those who are interested in how to calculate the table can contact me.
PROPOSITION 5 (other integrals). It is possible to calculate the two following integrals from the table. Those who are interested in how to calculate them can contact me.
PROPOSITION 6 (sum of normal random variables). Given the random vector X = () whose components are normally distributed, then the density of the random variable Y = is a normal law whose average and standard deviations are respectively given by:
Multivariate normal distribution
PROPOSITION 7. The joint probability density in the case of a random vector whose m components follow a normal distribution is:
Demonstration, first part. The aim of this proof will be to demonstrate that if we calculate the marginal distribution of from the given joint distribution, we obtain a normal distribution with an average given by . Moreover, we will prove that if we use this joint distribution to calculate the covariance between and , we obtain . We start operating the following change of variables:
whose Jacobian is the identity matrix. So we obtain for the joint density in Eq. 9 the expression:
We then consider the substitution
whose Jacobian is the determinant of P which is again the identity matrix, since P is an orthogonal matrix (P is the matrix introduced in Prop. 2, whose columns are eigenvectors of the covariance matrix). Then we have
And, according to Prop. 1, we obtain for the joint distribution in Eq. 9 the expression:
So, the marginal distribution of the first random variable is
We recognize the integrals in Prop. 4, for n = 0. So we have for the marginal distribution:
while the joint distribution becomes
Let’s now consider another change of variable, the following one:
whose Jacobian is given by:
Then, according to Prop. 1, we have
This proves that the variables are independent. But they are also normally distributed random variables whose average is zero and whose standard deviation is
for i that goes from 1 to m. Since we have
we can calculate the marginal distribution of ξ_j according to Prop. 6:
Remembering the very first substitution (Eq. 10) we then draw the following conclusion:
Now, if you remember Prop. 3, you can easily conclude that the marginal density of is, in fact, a normal distribution with average given by and standard deviation given by . This concludes the first part of the demonstration. It is worth noting that we have calculated, in the previous lines, a very complex integral (the first collected in the following paragraph), and we can be proud of ourselves.
Demonstration, second part. We have now to prove that the covariance coefficient of between and , is given by . In order to do that, we’ll calculate the covariance between and with the formula
To simplify the writing, let’s assume i=1 and j=2. For we have:
Now, considering again Prop. 4, we easily recognize that:
So, the integral becomes:
For we have:
So, is zero and the same applies to , as the reader can easily discover by herself, using Eq. 20. Hence, we have found:
Now, just consider Eq. 7 (the second one) in Prop. 3, and you will recognize that we have found
which is exactly what we were trying to demonstrate. The reader has likely realized that we have just calculated another complex integral, the second one in the following paragraph. It can be also verified that the joint density is, in fact, a density: in order for that to be true it must be
Now, if we use the substitutions in Eq. 10 and in Eq. 12 we obtain:
And our proof is now complete.
Prop. 7 is nothing more than the calculation of three very complex integrals. I have collected these results in what follows. Consider that you can substitute the covariance matrix with any symmetrical one, and these formulae still hold.
Let , , …, be independent random variables with an exponential distribution. We have already found the expression of the distribution of the random variable Y = + + …+ when , , …, have pairwise distinct parameters (here). The reader will easily recognize that the formula we found in that case has no meaning when the parameters are all equal to λ. I will derive here the law of Y in this circumstance. This can be done with a demonstration by induction, with no particular effort, but I will follow a longer proof. Why? Perhaps because I like pain? Yes, this might be true; but the main reason is that, to me, this longer demonstration is quite interesting and it gives us the chance to introduce the following proposition. In this blog post, we will use some of the results from the previous one on the same topic and we will follow the same enumeration for propositions.
PROPOSITION 8 (sum of m independent random variables). Let , , …, be independent random variables. The law of Y = + + …+ is given by:
Proof. We know that the thesis is true for m=2 (PROP. 1). Consider now that:
But we know that , , …, are independent. So we have:
We now operate the substitution
And we obtain:
But this has to be true for any possible interval [a, b], which means that:
This is one of the equations in the thesis. The other ones can be derived in the same way ♦
Guessing the solution
I will solve the problem for m = 2, 3, 4 in order to have an idea of what the general formula might look like.
PROPOSITION 9 (m=2). Let , be independent exponential random variables with the same parameter λ. The law of = + is given by:
for y > 0. It is zero otherwise.
Proof. We just have to substitute , in Prop. 1. We obtain:
And we find the thesis by solving the integral ♦
PROPOSITION 10 (m=3). Let , , be independent exponential random variables with the same parameter λ. The law of Y = + + is given by:
for y>0. It is zero otherwise.
Proof. If we define = + and = , then we can say – thanks to Prop. 2 – that and are independent. This means that – according to Prop. 1 – we have
And the proof is concluded ♦
PROPOSITION 11 (m=4). Let , , , be independent exponential random variables with the same parameter λ. The law of Y = + + + is given by:
Proof. We can demonstrate the thesis in the same way we did in Prop. 10. We can draw the same conclusion by directly applying Prop. 8:
The domain of integration is the tetrahedron in Figure 1. The generic point P within the tetrahedron has to belong to the segment AC that, in turns, belongs to the triangle shaded grey. This means that the domain of integration can be written [0, y]×[0, y – ]×[0, y – – ]. Thus, we calculate:
and the proof is concluded ♦
The reader has now likely guessed what the density of Y looks like when m is the generic integer number. But before we can rigorously demonstrate that formula, we need to calculate an integral.
PROPOSITION 12 (lemma). It is true that:
Proof. We have already found that the thesis is true when m = 4. Let us now assume that it is true for m – 1; if we write the thesis for the m – 2 variables , , … , we have:
Now we put η = y – :
By integrating both members in [0, y] we obtain:
and the proof is concluded ♦
PROPOSITION 13. Let , , …, be independent random variables. The law of Y = + + …+ is given by:
Proof. By directly applying Prop. 8 we have:
Thus, we can write:
The domain of integration can be written in a more proper way (as we did in Prop. 11) as follows:
But this is the integral calculated in Prop. 12, and the proof is concluded ♦
A numerical application
The reader might have recognized that the density of Y in Prop. 13 is, in fact, a Gamma distribution: it is the distribution Γ(m, λ), also known as Erlang’s distribution. There might even be a reader who perhaps remembers that I have discussed that distribution in a post of mine (here) and that I have coded a program that plots both the density and the distribution of such a law. In Figure 2 we see the density of Y (left) and the distribution function (right) for λ = 1.5 and for m that goes from 1 to 3 (note that m is indicated with α in that picture).
For the last four months, I have experienced the worst level of my illness: I have been completely unable to think for most of the time. So I could do nothing but hanging in there, waiting for a miracle, passing from one medication to the other, well aware that this state could have lasted for years, with no reasonable hope of receiving help from anyone. This has been the quality of my life for most of the last two decades. Then, some days ago, the miracle happened again and I found myself thinking about a theorem I was working on in July. And once more, with a great effort, my mind, which is not so young anymore, started her slow process of recovery. I concluded this proof last night. This is only a poor thing but since it is not present in my books of statistics, I have decided to write it down in my blog, for those who might be interested.
I can now come back to my awkward studies, which span from statistics to computational immunology, from analysis of genetic data to mathematical modelling of bacterial growth. Desperately searching for a cure.
Let be independent random variables with an exponential distribution with pairwise distinct parameters , respectively. Our problem is: what is the expression of the distribution of the random variable ? I faced the problem for m = 2, 3, 4. Then, when I was quite sure of the expression of the general formula of (the distribution of Y) I made my attempt to prove it inductively. But before starting, we need to mention two preliminary results that I won’t demonstrate since you can find these proofs in any book of statistics.
PROPOSITION 1. Let be independent random variables. The distribution of is given by:
where f_X is the distribution of the random vector .
PROPOSITION 2. Let be independent random variables. The two random variables and (with n<m) are independent.
DEFINITION 1. For those who might be wondering how the exponential distribution of a random variable with a parameter looks like, I remind that it is given by:
Guessing the solution
As mentioned, I solved the problem for m = 2, 3, 4 in order to understand what the general formula for might have looked like.
PROPOSITION 3 (m = 2). Let be independent exponential random variables with distinct parameters , respectively. The law of is given by:
Proof. We just have to substitute in Prop. 1. We obtain:
And the demonstration is complete ♦
PROPOSITION 4 (m = 3). Let be independent exponential random variables with pairwise distinct parameters , respectively. The law of is given by:
Proof. If we define and , then we can say – thanks to Prop. 2 – that and are independent. This means that – according to Prop. 1 – we have
The reader will now recognize that we know the expression of because of Prop. 3. So, we have:
For the first integral we find:
For the second one we have:
Hence, we find:
And the thesis is proved ♦
PROPOSITION 5 (m = 4). Let be independent exponential random variables with pairwise distinct parameters , respectively. The law of is given by:
for y>0, while it is zero otherwise.
Proof. Let’s consider the two random variables , . Prop. 2 tells us that are independent. This means that – according to Prop. 1 – we can write:
The reader has likely already realized that we have the expressions of and , thanks to Prop. 3. So we have:
For the four integrals we can easily calculate what follows:
Adding these four integrals together we obtain:
And this proves the thesis ♦
We are now quite confident in saying that the expression of for the generic value of m is given by:
for y>0, while being zero otherwise. But we aim at a rigorous proof of this expression.
In order to carry out our final demonstration, we need to prove a property that is linked to the matrix named after Vandermonde, that the reader who has followed me till this point will likely remember from his studies of linear algebra. The determinant of the Vandermonde matrix is given by:
PROPOSITION 6 (lemma). The following relationship is true:
Proof. In the following lines, we calculate the determinant of the matrix below, with respect to the second line. In the end, we will use the expression of the determinant of the Vandermonde matrix, mentioned above:
But this determinant has to be zero since the matrix has two identical lines, which proves the thesis ♦
PROPOSITION 7. Let be independent exponential random variables with pairwise distinct parameters , respectively. The law of is given by:
for y > 0, while being zero otherwise.
Proof. We already know that the thesis is true for m = 2, 3, 4. We now admit that it is true for m-1 and we demonstrate that this implies that the thesis is true for m (proof by induction). Let’s define the random variables and . These two random variables are independent (Prop. 2) so – according to Prop. 1 – we have:
Now, is the thesis for m-1 while is the exponential distribution with parameter . So we have:
For the sum we have:
The sum within brackets can be written as follows:
So far, we have found the following relationship:
In order for the thesis to be true, we just need to prove that
which is equivalent to the following:
Searching for a common denominator allows us to rewrite the sum above as follows:
So, we just need to prove that:
References. A paper on this same topic has been written by Markus Bibinger and it is available here.
In the following four tables, I present a collection of integrals and integral functions that are of common use in Statistics. The values for the Beta function have been calculated by a code of mine that uses – as a recursive function – another code that I wrote for the calculation of the Gamma Function (see the related blog post). Below these tables, you find a representation of the Beta function, plotted by my code. I have demonstrated each one of these results and these calculations will be included in an upcoming book.
In a previous blog post, I have discussed the case of normally distributed random variables. But what happens if we have a vector whose components are random variables with this kind of distribution? In particular, what is the expression of the joint probability density? I don’t know how to derive this formula from the marginal distributions, but I have been able to demonstrate, after several hours of work, the following proposition.
Proposition 1. The joint probability density in the case of a random vector whose two components follow a normal distribution is:
where σ_1 and σ_2 are the standard deviations, μ_1 and μ_2 are the expectations of the two random variables and ρ is the correlation coefficient between the two variables.
Proof. The marginal distribution of x_1 is given by
By operating the substitutions
This demonstrates that x_1 is in fact a normally distributed random variable and that its expectation and standard deviation are σ_1 and μ_1, respectively. The same applies to x_2. In order to demonstrate that ρ is the correlation coefficient between x_1 and x_2 we have to calculate the covariance of f_X, which is given by E[X_1,X_2]-E[X_1]E[X_2]. The first addend is given by:
Let’s consider the inner integral and the following substitutions
The two integrals inside the root square are not the easiest ones, and I will not discuss them here. Just consider that they can be solved using the well known results:
If we put these two integrals inside I, we obtain
The two inner integrals are two difficult ones, again. The first one can be solved using the same exponential integral used above:
For the other inner integral we have
The first and the second integrals can be solved using the table previously reported, while the third one is zero. At the end of all this work, we have:
And this proves that ρ is the correlation coefficient between x_1 and x_2. This is a complete demonstration of the expression of the joint probability density of a bivariate normal distribution ■
This formula can be extended to the case of m components.
Proposition 2. The joint probability density in the case of a random vector whose m components follow a normal distribution is:
where C is the covariance matrix, given by
Proof. This formula is easy to prove in the case of m=2, using proposition 1. I have no idea about how it could be proved when m>2 ■
I have written a code in Fortran (download) that asks you for expectations, standard deviations and correlation coefficient and plots the joint density of the bivariate normal distribution. I have written a code in Octave that does the same job (download). In the picture below you find the joint density for two standard normally distributed random variables with ρ=0 (left), ρ=0.7 (right, upper half) and ρ=-0.7 (right, lower half). These pictures have been plotted by the code in Fortran. The other image is a comparison between the two programs, for another bivariate normal distribution.
In what follows I will present a set of codes that I have written in Octave and in Fortran for the study of random variables that follow a gamma distribution. In another post, we will discuss the beta distribution.
2_The gamma distribution
A random variable is said to have a gamma distribution with parameters (α, λ) if its density is given by
with α, λ>0 and where Γ(α) is the gamma function, which is quite a pain in the lower back, as you will see. This random variable can be used to describe the probability that the duration of life of an entity (whether it is a mechanical component or a bacterium) depends on time. The gamma distribution allows to describe the following situations:
for α>1 the probability of failure of a mechanical component (or death of an organism) increases as the time passes by (increasing hazard rate);
for α=1 the probability of failure is constant (constant hazard rate);
for α<1 the failure is less likely as time passes by (decreasing hazard rate).
The hazard rate is the limit below:
where F_X(t) is the distribution function of the gamma distribution. I have not been able to find an analytic expression for the hazard rate of the gamma distribution, but I have been able to obtain it with the computer-aided calculation (see below). But in order to calculate F_X(t), we need to calculate the gamma function…
3_The gamma function
Γ(α) (the gamma function) is an integral function that cannot be calculated analytically:
It has to be calculated with numerical integration, but a problem is that Γ(α) is an indefinite integral. One way to face this problem is to stop the numerical integration when the addend is below a certain cut off (let’s say 0.0001). It is worth noting that an important feature of this function is that if you know its value in an interval such as [x, x+1], then you can derive Γ(α) for any other possible value of α. This is due to the recursive formula: Γ(α+1) = αΓ(α). That said, I have calculated Γ(α) in [1, 2] and then I have used these values to define Γ(α) in [0, 4] with the .exe file (written in Fortran) available here. Download it and put it in a folder. When you run it, it gives a .txt file with the values of Γ(α) in [0, 4] and two .mat files that can be used to build the plot of Γ(α) (see figure 1). Values of Γ(α) in [0, 2] are tabulated below, while those of you who like the very long list of numbers can find a huge table with values of the gamma function in [0, 4] at the end of this article.
4_The distribution function
The distribution function of the gamma density cannot be calculated analytically so, again, I had to make good use of my poor coding skills and I wrote a code in Octave (here, it uses the two .mat files generated by the code in Fortran mentioned above!) and one in Fortran (available here). In figure 2 (code in Octave) you can find the density and the distribution function for different values of α.
The code in Fortran, on the other hand, asks the user for the values of α and λ and gives a plot for the density and one for the distribution function. Moreover, it gives a .txt file with the values of the distribution function. Note that you can only place values for α that are below 25, otherwise you generate an overflow condition. I am not sure why you have overflow condition with a value of α that is below 34, but this is another story. Nevertheless, for those who are interested, Γ(n) = (n-1)! and the biggest factorial that can be handled by a computer in single precision mode (where 8 bits are used for the exponent) is 33!. So, we have a little mystery here… The reason why I have stuck with single precision is that DISLIN – the graphics library – seems to be unable to handle double precision numbers. If you are interested in the gamma function, the density and the distribution function for α = 20 and λ=2, then the code in Fortran gives for Γ(20) a value of 1.216451E+17 and the two plots in figure 3.
5_The hazard rate
As mentioned in paragraph 2, the hazard rate of the gamma distribution cannot be easily studied. I forgot to mention that it is relatively easy to prove that it is constant for α=1. But what happens for α>1 and for α<1? In order to answer, I have written a code (download) that plots it and it seems to be possible to say that the hazard rate increases for α>1 and decreases for α<1. This means that the gamma distribution is suitable for modeling each of the three kinds of patterns for the probability of failure as the time passes by. In figure 4 you find an example of the hazard rate for α<1 (left) and an example for α>1 (right).
A table for the gamma function in [0.01, 3.99] is reported below. In each column, the value of α is indicated on the left while the corresponding value for Γ(α) is indicated on the right.