1. ## Multivariate Distributions

1. )If the joint probability density of X and Y is given by
f(x,y)=24xy, if 0<x<1, 0<y<1, x+y<1
and 0 elsewhere

Find P(X+Y<1/2)

2.)Find the joint probability density of the two random variables X and Y whose joint distribution function is give by
F(x,y)=(1-e^(-x^2))(1-e^(-y^2)) if x>0, y>0
and 0 elsewhere

2. Originally Posted by Yan
1. )If the joint probability density of X and Y is given by
f(x,y)=24xy, if 0<x<1, 0<y<1, x+y<1
and 0 elsewhere

Find P(X+Y<1/2)
$
P(X+Y<1/2)=\int_{y=0}^{1/2} \int_{x=0}^{1/2-y} f(x,y) \ dx dy
$

.

3. Originally Posted by Yan

2.)Find the joint probability density of the two random variables X and Y whose joint distribution function is give by
F(x,y)=(1-e^(-x^2))(1-e^(-y^2)) if x>0, y>0
and 0 elsewhere
$
F(x,y)=\begin{cases}
\int_{u=0}^x \int_{v=0}^y f(u,v)\ dvdu, & x,y>0 \\
0, & \text{otherwise}
\end{cases}
$

As $F(x,y)$ is separable you can assume that $f(u,v)$ is separable (can be written as the product of two one variable functions each in one of the two variables)

So:

$f(u,v)=\begin{cases} h(u)g(v),& u,v>0\\0, & \text{otherwise}\end{cases}$

and:

$1-e^{-x^2}=\int_{u=0}^x h(u)\ du$

$1-e^{-y^2}=\int_{v=0}^y g(v)\ dv$

.