Example

Example
The joint density function of X and Y is given by
$f(x,y)=\left\{ \begin{array}{ll} 2e^{-x}e^{-2y}&0<x<\infty,0<y<\infty \\ \\ 0&\mbox{ otherwise } \end{array}\right .$
Compute (a) $P\{X>1,Y<1\}$,(b) $P\{X<Y\}$, and (c) P{X<a}.
Solutions:
(a)
$\begin{array}{rcl} P\{X>1,Y<1\}&=&\displaystyle\int_0^1\int_1^\infty 2e^{-x}e^{... ...=&e^{-1}\displaystyle\int_0^1 2e^{-2y}dy \\ \\ &=&e^{-1}(1-e^{-2}) \end{array}$

(b)

$\begin{array}{rcl} P\{X<Y\}&=&\displaystyle\int\int\limits_{\hspace{-0.5cm}(x,y... ...\ &=&1-\displaystyle\frac{2}{3} \\ \\ &=&\displaystyle\frac{1}{3} \end{array}$

(c)

$\begin{array}{rcl} P\{X<a\}&=&\displaystyle\int_0^a\int_0^\infty 2e^{-2y}e^{-x}... ...t_0^a e^{-x}dx \\ \\ &=&1-e^{-a} \qquad\rule[0.02em]{1.0mm}{1.5mm} \end{array}$

Example
The joint density of X and Y is given by
$f(x,y)=\left\{ \begin{array}{ll} e^{-(x+y)}&0<x<\infty, 0<y<\infty \\ \\ 0&\mbox{ otherwise } \end{array}\right .$
Find the density function of the random variable X/Y.
Solutions:
We start by computing the distribution function of X/Y. For a>0,
$\begin{array}{rcl} F_{X/Y}(a)&=&P\left\{\displaystyle\frac{X}{Y}\leq a\right\} ... ...\right ]\Big \vert _0^\infty \\ \\ &=&1-\displaystyle\frac{1}{a+1} \end{array}$