# Let X and Y be independent ran

*label*SAT

*timer*Asked: Mar 31st, 2016

**Question description**

Let X and Y be independent random variables, each having a geometric distribution with parameter p, so that P(X=x,Y=y)=P(X=x)P(Y=y) by independence = (1-p)x-1p(1-p)y-1p x=1,2,..... y=1,2...... Let U=min{X,Y}, and V=X-Y (a) What are the possible values u and v of U and V respectively?(b) Find the joint pmf of U and V; that is, find P(U=u, V=v). (Hint: You might find it easier to look at the case v≥1, v≥-1, and v=0 separately) (c) U and V are independent random variables if P(U=u, V=v)=P(U=u)P(V=v) for all u and v. Determine wether this is the case here.