Question

Question:Suppose that two random variables X and Y have the joint p.d.f.\(f\left( {x,y} \right) = \left\{ \begin{array}{l}k{x^2}{y^2}\,\,\,\,\,\,\,\,\,\,\,\,for\,{x^2} + {y^2} \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\). Compute the conditional p.d.f. of X given

Y = y for each y.

Short answer

Conditional pdf of x given y=y for each y is

\(\left\{ \begin{array}{l}1.5{x^2}{\left( {1 - {y^2}} \right)^{^{ - \frac{3}{2}}\,}}\,\,\,\,{\rm{for}}\, - {\left( {1 - {y^2}} \right)^{\frac{1}{2}}} < x < {\left( {1 - {y^2}} \right)^{\frac{1}{2}}}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{otherwise}}\end{array} \right.\,\,\)

Step-by-step solution

Given Information.

For the random variables x and y

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}k{x^2}{y^2}\,\,\,\,\,\,\,\,\,\,\,\,for\,{x^2} + {y^2} \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)

Computing the Marginal probability

The marginal pdf\({f_1}\)and\({f_2}\)of x and y is

\({f_1}\left( x \right) = \left\{ \begin{array}{l}{e^{ - x}}\,\,\,\,for\,x \ge 0\\0\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)and\({f_2}\left( y \right) = \left\{ \begin{array}{l}2{e^{ - 2y}}\,\,\,\,for\,y \ge 0\\0\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)

When we multiply \({f_1}\left( x \right)\)times \({f_2}\left( y \right)\)and compare the product to \(f\left( {x,y} \right)\)we get \(k = 2\)

Computing the conditional probability

Now to compute the conditional pdf of\({f_{x|y}}\left( {x|y} \right)\)

\(\begin{array}{l}{f_{X|Y}}\left( {x|y} \right) = \frac{{{f_{X|Y}}\left( {x|y} \right)}}{{{f_Y}\left( y \right)}}\\ = \left\{ \begin{array}{l}1.5{x^2}{\left( {1 - {y^2}} \right)^{^{ - \frac{3}{2}}\,}}\,\,\,\,for\, - {\left( {1 - {y^2}} \right)^{\frac{1}{2}}} < x < {\left( {1 - {y^2}} \right)^{\frac{1}{2}}}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\,\,\end{array}\)

Hence,

\({f_{X|Y}}\left( {x|y} \right) = = \left\{ \begin{array}{l}1.5{x^2}{\left( {1 - {y^2}} \right)^{^{ - \frac{3}{2}}\,}}\,\,\,\,for\, - {\left( {1 - {y^2}} \right)^{\frac{1}{2}}} < x < {\left( {1 - {y^2}} \right)^{\frac{1}{2}}}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\) is the Conditional pdf of x given y=y for each y.