I am trying to generate $M$ random numbers which are exponentially distributed and whose sum adds up to $N$ (for simplicity, $N=1$).

I found that the generated numbers are initially exponentially distributed. However, after re-scaling they become uniformly distributed. What is the reason for that? And is there a solution?

Here is the result:

enter image description here

Any suggestions would be greatly appreciated.

P.S. My code written in Matlab:

samples = 10000;
lambda = 1;
X = -log(rand(samples,2))/lambda;
X = X./sum(X,2); % re-scaling
  • 2
    $\begingroup$ Note that sum(X,2) computes the sum of rows. $\endgroup$ – user226604 May 25 at 15:36
  • 1
    $\begingroup$ There is no such thing as "random numbers which are exponentially distributed and whose sum adds up to N:" an exponential distribution assigns some probability to arbitrarily large numbers, whereas limiting the sum to $N$ eliminates that possibility. Could you clarify what you actually need to accomplish? $\endgroup$ – whuber May 25 at 21:23

[Answer revised in view of helpful comment from @whuber.]

If you know the exponential rate $\theta,$ then dividing by $M\theta$ will give you a total near $N=1.$ If you don't know $\theta$ and $M$ is sufficiently large that the $\theta$ is well approximated by the reciprocal of the sample mean (a random variable), then you can still come close.

In what follows, I assume $\theta$ is unknown and $M$ is large. Then I adjust by dividing by the sum.

In R:

x = rexp(10000);  y = x/sum(x)
[1] 1

hist(y, prob=T, ylim=c(0, 10000), col="skyblue2")
curve(dexp(x, 1/mean(y)), add=T, col="red", lwd=2, n = 10001)

enter image description here

Even with smaller $M = 100,$ the adjusted sample has sum $1$ and is nearly exponential.

x = rexp(100);  y = x/sum(x)
[1] 1
[1] 97.64598

ks.test(y, "pexp", 100)

        One-sample Kolmogorov-Smirnov test

data:  y
D = 0.084865, p-value = 0.4674
alternative hypothesis: two-sided

The binning is slightly inconsistent between the two histograms below because $Y = X/97.646,$ not $X/100.$

enter image description here

  • 1
    $\begingroup$ This answer is an illusion owing to the large sample size. Try it with, say, a sample of 3 rather than 10,000. You will have to repeat your experiment a few times, but it will quickly become apparent that the distribution you create is far from exponential. $\endgroup$ – whuber May 25 at 21:22
  • 1
    $\begingroup$ Sorry and thanks. Somehow I was focused on approximate results and M large. Accordingly, I made changes to my Answer. $\endgroup$ – BruceET May 25 at 23:07

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