# Thread: Problem on Random Variables

1. ## Problem on Random Variables

I'm working on the following problem for a probability class.

Let $X_1, X_2,...$ be independent random variables with a common continuous distribution function. Let $B$ be the $\omega$-set where $X_m(\omega)=X_n(\omega)$ for some pair $m,n$ of distinct integers, and show that $P(B)=0$.

Part of my problem is that this section we're studying on random variables introduces so many new terms that I haven't been able to see how they work together very well. I understand that $P(B)=0$ means that $X_m(\omega)\ne X_n(\omega)$ for any $m\ne n$ except on a set of probability (measure) 0, but am struggling mostly with the first sentence. I'm not asking for a full fledged solution to the problem, but I would be grateful for any assistance on understanding the interplay between the other terms involved and advice on how to start.

2. Hello,

You can try to find the pdf of $X_m-X_n$ (by first finding the pdf of $(Y,Z)=(X_m-X_n,X_n)$ thanks to a Jacobian transformation and then integrating)

Then, you'll get something like $g(y)=\int f(y+z)f(z) ~dz$ for the pdf of Y.

$P(Y=0)=\int_{\{0\}} \int f(y+z)f(z) ~dz ~dy$

Reverse the integration order (what's in the integral is positive so we can apply Fubini-Tonelli's theorem) and you'll get 0, because you integrate over a set of Lebesgue measure 0.

Otherwise, maybe you can have a look into the properties of convolution... But right now I have to go to school... Good luck !