Question

Find E(X)

Answer 1

A joint probability distribution for random variables X and Y is:\[f(x,y) = \frac{3}{4}\left(\frac{1}{4^{x+y}} \right)\]
Find E(X) and E(Y)

Answer 2

That's for x=0, 1, 2, ... and y=0, 1, 2, ...

Answer 3

This doesn't look like a PDF to me...
\[\sum_{x=0}^\infty\sum_{y=0}^\infty \frac{3}{4^{x+y+1}}=\frac{3}{4}\sum_{x=0}^\infty\frac{1}{4^x}\sum_{y=0}^\infty\frac{1}{4^y}=\frac{3}{4}\times\frac{1}{1-\dfrac{1}{4}}\times\frac{1}{1-\dfrac{1}{4}}=\frac{4}{3}\neq1\]

Answer 4

Maybe you mean \(\dfrac{9}{16}\) instead of \(\dfrac{3}{4}\)?