Question

The joint probability distribution of X and Y is given by

f(x,y)=x+y, 0<x<1 and 0<y<1 zero otherwise.

Determine the joint distribution function of X and Y i.e F(x,y). Hence or otherwise calculate p(0<x<1/2,1/2<y<1)

Answer 1

joint distribution function:

$F(x,y)=P(X\le x,Y\le y)$

for $x\le 0$ or $y\le 0$ :

$F(x,y)=0$

for $x\ge 1$ and $y\ge 1$ :

$F(x,y)=1$

for $0<x<1 , 0<y<1$ :

$F(x,y)=\int^y_0\int^x_0f(u,v)dudv=\int^y_0\int^x_0(u+v)dudv=$

$=\int^y_0(x^2/2+vx)dv=yx^2/2+y^2x/2$

for $0<x<1$ and $x\ge 1$ :

$F(x,y)=F(x,1)=x^2/2+x/2$

for $0<y<1$ and $y\ge 1$ :

$F(x,y)=F(1,y)=y^2/2+y/2$

$P(0<x<1/2,1/2<y<1)=\int^1_{1/2}\int^{1/2}_0f(x,y)dxdy=$

$=yx^2/2+y^2x/2|^{1/2,1}_{0,1/2}=1/8+1/2=5/8$