Let $X$ and $Y$ be independent random variables. Each of them has a geometric distribution with $E[X] = 2$ and $E[Y] = 3$.
(a) Find the joint p.m.f. of $X$ and $Y$.
(b) Compute the probability that $X + Y \le 4$.
(c) Define two new random variables by $W = \min{ \left( X, Y \right ) }$ and $Z = \max{ \left( X, Y \right ) }$. Find the joint p.m.f. of $W$ and $Z$.
Ans:
$X$ and $Y$ are independent random variables. Each of them has a geometric distribution with $E[X] = 2$ and $E[Y] = 3$.
So joint p.m.f. of $X$ and $Y$,
$E[XY] = E[X] E[Y] = 2 * 3 = 6$
Is this correct?
And for $(b)$ and $(c)$ can anyone suggest how to proceed??
for (c), s min(X; Y ) is a geometric random variable with parameter 1 - (1 - p)^2 = 2p - p^2