91Ó°ÊÓ

Pdf of two random variables, X and Y is

\(f\left( {{\rm{x,y}}} \right) = \left\{ \begin{aligned}{l}c\sin x\;\;\;\;\;for\;0 \le x \le \frac{\pi }{2}\;\;and\;0 \le y \le 3\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;Otherwise\end{aligned} \right.\)

\(\begin{aligned}{f_1}\left( x \right) = \int\limits_y {f\left( {x,y} \right)dy} \\ = \int\limits_{y = 0}^3 {c\sin xdy} \end{aligned}\)

\(\begin{aligned}{f_1}\left( x \right) = c\sin x\int\limits_{y = 0}^3 {dy} \\ = 3c\sin x\end{aligned}\)

Therefore \({f_1}\left( x \right) = \left\{ \begin{aligned}{l}3c\sin x,\;\;\;\;0 \le x \le \frac{\pi }{2}\\0\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{aligned} \right.\)

Conditional pdf is given by:

\(\begin{aligned}{g_2}\left( {y|x} \right) = \frac{{f\left( {x,y} \right)}}{{{f_1}\left( x \right)}}\\ = \frac{{c\sin x}}{{3c\sin x}}\end{aligned}\)

\(\begin{aligned}{g_2}\left( {y|x} \right) = \frac{1}{{3\left( 1 \right)}}\\ = \frac{1}{3}\end{aligned}\)

\({g_2}\left( {y|x} \right) = \left\{ \begin{aligned}{l}\frac{1}{3},\;0 \le y \le 3\\0\;\;otherwise\end{aligned} \right.\)

Calculate the conditional pdf of Y given\(X = 0.73\)from pat a.

\({g_2}\left( {y|0.73} \right) = \left\{ \begin{aligned}{l}\frac{1}{3},\;0 \le y \le 3\\0\;\;\;otherwise\end{aligned} \right.\)

Computing the conditional probability

\(\begin{aligned}{\rm P}\left( {1 < y < 2|X = 0.73} \right) = \int\limits_{y = 1}^2 {{g_2}\left( {y|0.73} \right)} dy\\ = \int\limits_{y = 1}^2 {\frac{1}{3}} dy\end{aligned}\)

\(\begin{aligned}{\rm P}\left( {1 < y < 2|X = 0.73} \right) = \frac{1}{3}\left( y \right)_{y = 1}^2\\ = \frac{1}{3}\left( {2 - 1} \right)\\ = \frac{1}{3}\end{aligned}\)

therefore\({\rm P}\left( {1 < y < 2|X = 0.73} \right) = \frac{1}{3}\)