Question

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

 

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 independent?

Answer 1

(a) $f_x(x)=\int f_{x,y}(x,y)dy=\int_x^2\frac{1}{2}dy=\{ \frac{2-x}{2},\ \ 0\le x\le 2 \ and \ 0,\ \ else$

$f_y(y)=\int f_{x,y}(x,y)dx=\int_0^y\frac{1}{2}dx=\{\frac{y}{2},0\le y\le 2\ \ and \ \ 0, \ else$

(b) $f_{\frac{X}{Y}}(X/Y)=\dfrac{f_{x,y}(x,y)}{f_y(y)}=\dfrac{1/2}{y/x}=\{\frac{1}{y}, \ \ 0\le x\le y\ \ and\ \ 0,\ \ \ else$

$f_{Y/X}(Y/X)=\dfrac{f_{x,y}(x,y)}{f_x(x)}=\dfrac{1/2}{(2-x)/2}=\{\frac{1}{2-x},\ \ x\le y\le 2\ \ \ and \ 0,\ \ else$

(c) $E(X/Y=1)=\int_0^1f_{X/Y}(x/1)dx=\int_0^11dx=1$

(d) $f_{X/Y}(x/y)\neq f_x(x)\\$

$\Rightarrow X, Y$ are not statistically independent.