# Identifying a distribution

I am trying to understand what sort of distribution is produced by the following code.

Using the following Matlab code we can generate a distribution (normalized such that the sum of the set is equal to 1):

M=500; % Number of samples
z=5;
SUM = 1;
ns = rand(1,M).^z; % Uniformly distributed random numbers to the power of z
TOT = sum(ns);
X = (ns/TOT)*SUM; % Re-scaling
hist(X(1,:),100)


For the exponent $$z=1$$ the sampling distribution is essentially "flat", and it also has a small range. Enlarging $$z$$ gives a wider range of numbers. Here is an example:

I would like to classify this distribution and understand the underlying mathematics.

Any explanations would be greatly appreciated.

• Similar method, different distribution. Not sure that makes it a duplicate. – BruceET Jun 18 at 17:30
• @AdamO the question you link to is about $e^X$ where $X \sim \text{Uniform}(0, 1)$ while this question is asking about $X^z$. This leads to a different distribution. – olooney Jun 18 at 17:35

I will show this for $$U^2,$$ instead of $$U^5.$$ See @olooney's comment below my Answer.

Short answer: If $$U \sim \mathsf{Unif}(0,1),$$ then $$X = U^2 \sim \mathsf{Beta}(\frac 1 2, 1).$$

Proof: For $$x \in (0,1),$$ we have $$F_X(x) = P(X \le x) = P(U^2 \le x) = P(U \le x^{1/2}) = x^{1/2}.$$ Then $$f_X(x) = F_X^\prime(x)$$ is easily seen to be the density of $$\mathsf{Beta}(\frac 12, 1).$$ See Wikipedia on beta distributions.

Demonstration by simulation:

set.seed(618)
u = runif(10^6);  x = u^2
mean(x)
[1] 0.3330528     # aprx E(X) = 1/3 from simulation
.5/(.5+1)
[1] 0.3333333     # exact E(X) from formula

cutp.u = seq(0, 1, by = 0.1);  cutp.x = cutp.u^2
frb = rainbow(12)
par(mfrow = c(1,3))
hist(u, prob=T, br=cutp.u, col=frb, ylim=c(0,10), main="UNIF(0,1)")
curve(dunif(x), 0, 1, add=T, n=10001, lwd=2)
hist(x, prob=T, br=cutp.x, col=frb, main="BETA(.5,1)")

• Very nice! However, because the question concerns the $z=5$ power rather than the square, it would be better to generalize your answer a little. – whuber Jun 18 at 17:34
• Generalizes to $X^z \sim \text{Beta}(1/z, 1)$, where $X \sim \text{Uniform}(0, 1)$. – olooney Jun 18 at 17:40
• Both absolutely correct. I'm still fussing with this. Will edit accordingly: (a) Not sure if it's homework, so want to show method without just being an answerbook. (b) Difficult to show clear graphs for $z = 5.$ – BruceET Jun 18 at 17:45
• It seems nobody (including the OP) cares about the summation performed after the transformation. The resulting RVs are of the form $$\frac{X_i}{\sum_i X_i}$$ – gunes Jun 19 at 0:54