Sampling distribution's magnitude variation

[roam]

Homework Statement

What is the name of the following distribution, and why does increasing the exponent increase the magnitude variation?

Relevant Equations

Matlab code is shown below.

I have generated a distribution (normalized such that the sum is equal to 1) by using the code:

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

For an exponent $z=1$, the sampling distribution is "flat", but it also has a shorter range. Enlarging $z$ gives a wider range of numbers. Why?

Here is a comparison of the results for $z=1$ and $z=5$:

Also, is it possible to have both a wide range and a "flat" distribution at the same time (i.e. very large numbers being just as likely as the small numbers)?

Explanations would be greatly appreciated.

 

[Stephen Tashi]

Problem Statement: What is the name of the following distribution, and why does increasing the exponent increase the magnitude variation?

For an exponent $z=1$, the sampling distribution is "flat", but it also has a shorter range.

You seem to be saying that you are plotting a histogram of 500 samples from a single random variable. If that is the claim, what random variable is it?

By way of analogy, suppose I take a sample of two numbers as follows. I pick a person at random and measure his height $H$ in inches and his weight $W$ in pounds. I form the two numbers $X_1 = H/ (H+W)$ and $X_2 = W/(H+W)$. Can I regard $X_1$ and $X_2$ as two independent samples of the same random variable? What would that single random variable be?

To extend this, suppose I pick a person and measure 500 different aspects of that person that have numerical measures - height, weight, minutes spent sleeping today, ounces of water consumed today, ...etc. I take the sum $TOT$ of all the measurements and form a set of 500 numbers by dividing the original measurements by $TOT$, obtaining the values $X_1, X_2, ...X_{500}$. Then I histogram these values and regard it as 500 independent measures of the same random variable $X$. How can I define a random variable $X$ so that would make sense?

I'm not saying it's mathematically absurd to make an attempt to define $X$. It may be absurd from the point of view of practical applications, but it's a very interesting mathematical question. If we have 500 distinct random variables with some given joint distribution then under what conditions can we define a single random variable $X$ such that 500 independent samples of $X$ can be realized by treating 1 sample from the joint distribution of the 500 different random variables as if they were 500 independent measurements of the same random variable?

In the case where the 500 jointly distributed random variables are independent and identically distributed then we can obviously treat 1 sample of the 500 values as 500 independent realizations of the random variable $X$ , where $X$ has the distribution common to each of the 500 random variables.

However, in your case, you aren't dealing with 500 independently distributed values, since outcome of each value depends on $TOT$, which is determined jointly from 500 other values.

Unless we can define a single random variable $X$ whose probability density is approximated by your histograms, attempts to answer your question by thinking about the probability density of a single random variable are heading down the wrong track.