# Independence of Random Variables

## Homework Statement

Suppose X is a discrete random variable with probability mass function
pX(x)=1/5, if x=-2,-1,0,1,2
pX(x)=0, otherwise
Let Y=X2. Are X and Y independent? Prove using definitions and theorems.

## The Attempt at a Solution

The random variables X and Y are independent <=> pX,Y(x,y)=pX(x)pY(y) for ALL x,y E R

But the trouble here is that we don't have pX,Y(x,y) and pY(y). What can we do?

Thanks for any help!

Related Calculus and Beyond Homework Help News on Phys.org
You will first have to ccalculate pY and pX,Y.
For example
$$p_Y(1)=P(Y=1)=P(X=1\vee X=-1)=P(X=1)+P(X=-1)=1/5+1/5=2/5$$
IN the same way calculate pY(y) for the remaining values in the range of Y, and similarly for pX,Y. Of course you would suspect X and Y not to be independent, so it would suffice to find one case where the joint distribution does not factorize.

How can I find the joint mass function pX,Y from their marginal mass functions?

$$p_{X,Y}(1,2)=P(X=1\vee Y=2)=P(X=1\vee X^2=2)...?$$
Which X values contribute to this probabbility. They should be such that X=1 and X^2=2

$$p_{X,Y}(1,2)=P(X=1\vee Y=2)=P(X=1\vee X^2=2)...?$$
Which X values contribute to this probabbility. They should be such that X=1 and X^2=2
How is this possible? Is the probability 0?
What other cases do I have to do?

How is this possible? Is the probability 0?
What other cases do I have to do?
Yes, the prob. is zero. Either do all remaining cases or just compare
$$p_{X,Y}(1,2)=0$$
to
$$p_{X}(1)p_{Y}(2)=?$$