Problem on Random Variables

Diamondlance · #1 $Old$ October 25th, 2009, 07:09 PM

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.

Moo · #2 $Old$ October 26th, 2009, 01:46 AM

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 !