Question

(1 point) Suppose that for two random variables X and Y the joint density function is f(x,y)=6xe−x(y+6), for x>0 and y>0. Find each of the following. (a) fX|Y(x,y)= (b) fY|X(x,y)=

Answer 1

First find the marginal distributions:

$f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_0^\infty6e^{-x(y+6)}\,\mathrm dy$

$\implies f_X(x)=\begin{cases}6e^{-6x}&\text{for }x>0\\0&\text{otherwise}\end{cases}$

$f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\int_0^\infty6e^{-x(y+6)}\,\mathrm dx$

$\implies f_Y(y)=\begin{cases}\dfrac6{(y+6)^2}&\text{for }y>0\\0&\text{otherwise}\end{cases}$

Then to get each conditional distribution, divide the joint distribution by the corresponding marginal distribution.

a.

$f_{X|Y}(x,y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}=\begin{cases}x(y+6)^2e^{-x(y+6)}&\text{for }x>0\\0&\text{otherwise}\end{cases}$

b.

$f_{Y|X}(x,y)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}=\begin{cases}xe^{-xy}&\text{for }y>0\\0&\text{otherwise}\end{cases}$