Re-scaling of exponentially distributed numbers

[roam]

TL;DR

I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$. However, the re-scaling always causes the numbers to become uniformly distributed.

For simplicity, let $N=1$. The following histograms show my results. The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed.

What is the cause of that, and is there a solution?

P.S. Here is my code in Matlab:

subplot(121)
samples = 10000;
lambda = 1;
X = -log(rand(samples,2))/lambda;
hist(X(:,1),100)
subplot(122)
X = X./sum(X,2); % re-scaling
hist(X(:,1),100)

 

[phinds]

The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed.

Yes, of course they do. That's the way math works. If you have a log distribution on a log scale, that's the same as a flat distribution in a linear scale. I don't know why you would expect otherwise. There IS no "solution".

 

[Stephen Tashi]

Summary: However, the re-scaling always causes the numbers to become uniformly distributed.

The mathematical question is more likely to be answered if it is stated precisely. As I read the MATLAB code, the mathematical question is:

$X$ and $Y$ are independent random variables and each is uniformly distributed on [0,1]. What is the distribution of $W = \frac{ \log(X)}{ \log(X) + \log(Y)}$ ?

 

[FactChecker]

I don't quite understand why you are scaling by sum(X,2). That will rescale each row by the sum of the numbers on that row (2 numbers). I changed it to sum(X,1), which sums the 10000 numbers along index 1 (sums each column), and got what I think you were expecting. I like to keep it simple and see the intermediate calculations so that I can make sure it is doing what I expected:
N=10
S = sum(X,1)
Y = N*X(:,1)/S(1,1); % re-scaling

 

[pbuk]

Summary: I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$.

If their sum adds up to a given N then their distributions are not independent and so they cannot individually be exponentially distributed. Take the case of M = 2; If the first number $n_0$ is exponentially distributed in the range $[0,N]$ but the second number must always equal $N-n_0$.

However, the re-scaling always causes the numbers to become uniformly distributed.

Yes of course, because as @StephenTashi points out your 'rescaling' creates a completely different distribution.

 

[FactChecker]

If their sum adds up to a given N then their distributions are not independent

He was trying to generate the data and do a simple linear rescaling of the data afterward. It should not have changed the general shape of the distribution. He had a MATLAB coding error.

 

[Stephen Tashi]

He was trying to generate the data and do a simple linear rescaling of the data afterward. It should not have changed the general shape of the distribution.

As to rescaling, I think the goal is take pairs of random variables $X_a, X_b$ , and from each pair , create the pair $W_a = X_a/(X_a + X_b),\ W_b = (X_b)/(X_a + X_b)$. Then we look at the distribution of $W_a$. So the intent is not to do a linear rescaling. The intent is to create pairs of random variables $W_a,\ W_b$ that sum to 1.

 

[FactChecker]

As to rescaling, I think the goal is take pairs of random variables $X_a, X_b$ , and from each pair , create the pair $W_a = X_a/(X_a + X_b),\ W_b = (X_b)/(X_a + X_b)$. Then we look at the distribution of $W_a$. So the intent is not to do a linear rescaling. The intent is to create pairs of random variables $W_a,\ W_b$ that sum to 1.

That is what his original code did. I don't know what the real intention was. He got a valid answer to either case in this thread. I didn't see anything about "pairs" in the description. I still think that my assumption fits the original description better (or at least as well).

 

[Stephen Tashi]

I don't know what the real intention was. He got a valid answer to either case in this thread.

We don't yet have a good mathematical explanation for the shape of the second histogram. I agree that we don't yet have a clear statement of a mathematical question!

Summary: I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$.

The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed.

It's often hard to translate a procedure into a question about random variables. The first step is to describe the procedure clearly.

As a procedure, one interpretation of what you want to do, in general, is to generate 1 set of $M$ random numbers that sum to $N$ and then you want to make a histogram of all those $M$ random numbers. - i.e. all $M$ of the numbers contribute to the histogram. Your claim is that most such histograms are approximately uniform distributions.

A different interpretation is that you want to generate many sets of $M$ random numbers, each set being one where the $M$ numbers sum to the same $N$. Then you want to make a histogram by using one number from each of the sets. For example, if you generate 100 sets of $M= 20$ numbers, you might make a histogram by picking the first number from each of the 100 sets of numbers. The histogram would involve 100 values.

Or, to make a hybrid of the previous procedures, perhaps you want to generate, say, $100$ sets of $M = 20$ random numbers such that the sum of the numbers in each set is $N$. Then you want to histogram all 2000 of the numbers.