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Chapter 3: Q3.6-5E (page 151)

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

Expert verified
  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

01

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

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

02

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

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

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Most popular questions from this chapter

Suppose that a coin is tossed repeatedly in such a way that heads and tails are equally likely to appear on any given toss and that all tosses are independent, with the following exception: Whenever either three heads or three tails have been obtained on three successive tosses, then the outcome of the next toss is always of the opposite type. At time\(n\left( {n \ge 3} \right)\)let the state of this process be specified by the outcomes on tosses\(n - 2\),\(n - 1\)and n. Show that this process is a Markov chain with stationary transition probabilities and construct the transition matrix.

Suppose that a Markov chain has four states 1, 2, 3, 4, and stationary transition probabilities as specified by the following transition matrix

\(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{4}}&{\frac{1}{4}}&0&{\frac{1}{2}}\\0&1&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}&0\\{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}\end{array}} \right]\):

a.If the chain is in state 3 at a given timen, what is the probability that it will be in state 2 at timen+2?

b.If the chain is in state 1 at a given timen, what is the probability it will be in state 3 at timen+3?

Suppose that the p.d.f. of X is as given in Exercise 3.

Determine the p.d.f. of \(Y = 3X + 2\)

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}cx\;\;for\;x = 1,...,5,\\0\;\;\;\;otherwise\end{array} \right.\)

Determine the value of the constantc.

Question: Suppose that the joint p.d.f. of two random variablesXandYis as follows:

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

Determine (a) the value of the constantc;

\(\begin{array}{l}\left( {\bf{b}} \right)\,{\bf{Pr}}\left( {{\bf{0}} \le {\bf{X}} \le {\bf{1/2}}} \right){\bf{;}}\,\left( {\bf{c}} \right)\,{\bf{Pr}}\left( {{\bf{Y}} \le {\bf{X + 1}}} \right)\\\left( {\bf{d}} \right)\,{\bf{Pr}}\left( {{\bf{Y = }}{{\bf{X}}^{\bf{2}}}} \right)\end{array}\)
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