The joint cdf of a bivariate r.v (X,Y) is given by

$F_{XY}$ (x,y)=(1-$e^{-ax} $)(1-$e^{-βy}$ ) x≥0,y≥0,α,β>0

=0 otherwise

• Find the marginal cdf’s of X & Y
• Show that X and Y are independent
• Find P(X≤1,Y≤1),P(X≤1),P(Y>1)& P(X>x,Y>y)

**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

Using the following properties the Marginal cdf of X and Y is given as :

F(x,∞)= $F_X$ (x)

F(∞,y)=$F_Y$ (y)

$F_X$ (x)=F(x,∞)=(1-$e^{-ax}$ )(1-0)=(1-$e^{-ax} $)

$F_Y$ (y)=F(∞,y)=(1-0)(1-$e^{-βy}$ )=(1-$e^{-βy}$ )

To show X and Y are independent i.e $F_{XY}$ (x,y)=$F_X$ (x).$F_Y$ (y)

L.H.S: $F_{XY}$ (x,y)=(1-$e^{-ax}$ )(1-$e^{-βy}$ )

R.H.S: $F_X$ (x).$F_Y$ (y)= (1-$e^{-ax}$ )(1-$e^{-βy}$ )

Hence $F_{XY}$ (x,y)=$F_X$ (x).$F_Y$ (y)