Question

13. Let X,Y be rv's with pd.f. f given byf(x,y)-'e, x>0. y>o. Calculate the following quantities

Answer 1

13)

X and Y are independent exponentially distributed

as

f(x,y) = f(x) * f(y)

f(x) = $\lambda e^{- \lambda x}$

f(y) = $\lambda e^{- \lambda y}$

$E(X) = \frac{1}{\lambda }$

$\sigma ^2{(X)} = \frac{1}{\lambda^2 }$

$E{(Y)} = \frac{1}{\lambda }$

$\sigma ^2{(Y)} = \frac{1}{\lambda^2 }$

Since X and Y are independent

$E(X|Y = y) = \frac{1}{\lambda }$

$\sigma ^2{(X|Y = y)} = \frac{1}{\lambda^2 }$