Question

Suppose that the joint p.d.f. of two points X and Y chosen by the process described in Example 3.6.10 is as given by Eq. (3.6.15). Determine (a) the conditional p.d.f.of X for every given value of Y , and (b)\({\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{4}} \right)\)

Short answer

1. Conditional pdf of X for every fiven value of Y is\(\frac{{ - 1}}{{\left( {1 - x} \right)\log \left( {1 - y} \right)}}\)
2. \({\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{4}} \right)\) is\(\frac{1}{2}\)

Step-by-step solution

Calculating Conditional pdf

a)

Joint pdf of X, Y will be:

\(f\left( {x,y} \right) = \left\{ \begin{aligned}{l}\frac{1}{{1 - x}}dx\,\,0 < x < y < 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{aligned} \right.\)

Hence, for 0<y<1 and 0<x<y

\(\begin{aligned}{g_1}\left( {x|y} \right) = \frac{{f\left( {x,y} \right)}}{{{f_2}\left( y \right)}}\\ = \frac{{ - 1}}{{\left( {1 - x} \right)\log \left( {1 - y} \right)}}\end{aligned}\)

Calculating probabilities

b)

Computing the probability for \(\left( {X > \frac{1}{2}|Y = \frac{3}{4}} \right)\):

When\(Y = \frac{3}{4}\)it follows part (a)

\({g_1}\left( {x|y = \frac{3}{4}} \right) = \left\{ \begin{aligned}{l}\frac{1}{{\left( {1 - x} \right)\log 4}}\,\,\,\,\,\,\,\,for0 < x < \frac{3}{4}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{aligned} \right.\)

Therefore

\(\begin{aligned}{\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{4}} \right) = \int\limits_{\frac{1}{2}}^{\frac{3}{4}} {{g_1}\left( {x|y = \frac{3}{4}} \right)} dx\\ = \frac{{\log 4 - \log 2}}{{\log 4}}\\ = \frac{1}{2}\end{aligned}\)