Chapter 5.1, Problem 16E

a.

To determine

Find the value of c such that the function $f\left(x,y\right)=c{e}^{-2x-3y}$ for $x>0$ and $0<y<x$ satisfies the properties of a joint probability density function.

Compute $P\left(X<1,Y<2\right)$.

b.

To determine

Compute $P\left(1<X<2\right)$.

c.

To determine

Compute $P\left(Y>3\right)$.

d.

To determine

Compute $P\left(X<2,Y<2\right)$.

e.

To determine

Compute $E\left(X\right)$.

f.

To determine

Compute $E\left(Y\right)$.

g.

To determine

Find marginal probability distribution of X.

h.

To determine

Find the conditional probability distribution of Y given that $X=1$.

i.

To determine

Find $E\left(Y|X=1\right)$.

j.

To determine

Find the conditional probability distribution of X given that $Y=2$.