Proving X1X2 ~ Y1Y2: Distribution of X1 and Y1 is the Same as X2 and Y2

[shan]

Homework Statement

If X1 ~ Y1 and X2 ~ Y2, then X1X2 ~ Y1Y2, prove or find a counterexample. (the distribution of X1 has the same distribution of Y1 etc)

Homework Equations

-

The Attempt at a Solution

I'm guessing the statement is true. For example if X1 and Y1 were both uniform and X2 and Y2 are binomial, then a uniform * binomial is distributed the same as a uniform * binomial.

 

[HallsofIvy]

It would help if you also told us what "~" meant.

 

[shan]

Ah sorry, it's the stuff I put in the brackets ie.

X1 ~ Y1 : the distribution of X1 has the same distribution of Y1, X1 and Y1 being random variables.

The same with X2 ~ Y2 and X1X2 ~ Y1Y2.

 

[shan]

If anyone was interested, the answer was no. The counter example was:

Let P(X_1 = 1) = P(X_1 = -1) = 0.5. Let X_2 = -X_1 and Y_1 = Y_2 = X_1. X_1 X_2 = -X_1^2 = -1 almost surely, but Y_1 Y_2 = X_1^2 = 1