Question

Q. 1. joint pdf of random variables ‘X’ and ‘Y’ is

F(x,y)=1/2, 0 <x<y, 0<y<2

 

Find

a. The marginal pdfs, f_X(x) and f_Y(y).

b. The conditional pdfs, f_X/Y(x/y) and f_Y/X(y/x)

c. The E(X/Y=1)

d. Are ‘X’ and ‘Y’ statistically The independent?

Answer 1

$F(x,y) = \dfrac{1}{2} , 0 <x<1, 0<y<2$

a.) $f_X(x) = \int_{-\infty}^{\infty}f_{XY}(x,y)dy$

$= \int_{0}^{2}\dfrac{1}{2}dy = 1$

$f_Y(y) = \int_{0}^{1}\dfrac{1}{2}dx = \dfrac{1}{2}$

b.) $fX/Y(x/y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y} = \dfrac{1}{2}$

$f_{Y|X}(y|x) = \dfrac{f_{X,Y}}{f_X(x)} = 1$

c.) $E(X/Y=1) = \int_{-\infty}^{\infty}xf_{X|Y}(x|y)dx = \int_{0}^{1}x.\dfrac{1}{2}dx = \dfrac{1}{4}$

d.) Since, $f_X(x) \ne f_Y(y)$

Hence,X and Y are statistically independent.