Independence of Random Variables

1. Dec 6, 2008

kingwinner

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
3. 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!

2. Dec 6, 2008

Pere Callahan

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.

3. Dec 6, 2008

kingwinner

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

4. Dec 7, 2008

Pere Callahan

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

5. Dec 7, 2008

kingwinner

How is this possible? Is the probability 0?
What other cases do I have to do?

6. Dec 7, 2008

Pere Callahan

Yes, the prob. is zero. Either do all remaining cases or just compare
$$p_{X,Y}(1,2)=0$$
to
$$p_{X}(1)p_{Y}(2)=?$$