Chapter 15.4, Problem 29E

Suppose X and Y are random variables with joint density function

$f\left(x,y\right)=\{\begin{array}{l}0.1e^{-\left(0.5x+0.2y\right)}\text{ if }x\ge 0,y\ge 0\\ 0\text{ otherwise}\end{array}$

(a) Verify that f is indeed a joint density function.

(b) Find the following probabilities.

(i) $P\left(Y\ge 1\right)$

(ii) $P\left(X\le 2,Y\le 4\right)$

(c) Find the expected values of X and Y.