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Chapter 3: Q 12E (page 167)

Let X be a random vector that is split into three parts,\(X = \left( {Y,Z,W} \right)\)Suppose that X has a continuous joint distribution with p.d.f.\(f\left( {y,z,w} \right)\).Let\({g_1}\left( {y,z|w} \right)\)be the conditional p.d.f. of (Y, Z) given W = w, and let\({g_2}\left( {y|w} \right)\)be the conditional p.d.f. of Y given W = w. Prove that\({g_2}\left( {y|w} \right) = \int {{g_1}\left( {y,z|w} \right)dz} \)

Short Answer

Expert verified

\({g_2}\left( {y|w} \right) = \int {{g_1}\left( {y,z|w} \right)dz} \)

Step by step solution

01

Given information

A random vector X is split into three parts.\(X = \left( {Y,Z,W} \right)\)

X has continuous joint pdf\(f\left( {y,z,w} \right)\)

02

compute the probability

Marginal Joint pdf of Y and W =\(h\left( {y,w} \right)\)

Marginal pdf of w be\({h_2}\left( w \right)\)

Therefore

\(h\left( {y,w} \right) = \int {f\left( {y,z,w} \right)dz,} \)

\({h_2}\left( w \right) = \int {\int {f\left( {y,z,w} \right)\,dzdy} } \)

\({g_1}\left( {y,z|w} \right) = \frac{{f\left( {y,z,w} \right)}}{{{h_2}\left( w \right)}}\)

\({g_2}\left( {y|w} \right) = \frac{{h\left( {y,w} \right)}}{{{h_2}\left( w \right)}}\)

\(\begin{align}{g_2}\left( {y|w} \right) &= \frac{{\int {f\left( {y,z,w} \right)dz} }}{{{h_2}\left( w \right)}}\\ &= \int {{g_1}\left( {y,z|w} \right)} dz\end{align}\)

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Most popular questions from this chapter

Each student in a certain high school was classified according to her year in school (freshman, sophomore, junior, or senior) and according to the number of times that she had visited a certain museum (never, once, or more than once). The proportions of students in the various classifications are given in the following table:

Never once More than once

than once

Freshmen 0.08 0.10 0.04

Sophomores 0.04 0.10 0.04

Juniors 0.04 0.20 0.09

Seniors 0.02 0.15 0.10

a. If a student selected at random from the high school is a junior, what is the probability that she has never visited the museum?

b. If a student selected at random from the high school has visited the museum three times, what is the probability that she is a senior?

Return to the situation described in Example 3.7.18. Let\(x = \left( {{x_1},.....,{x_5}} \right)\)and compute the conditional pdf of Z given X = x directly in one step, as if all of X were observed at the same time.

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.

Return to Example 3.10.13. Prove that the stationary distributions described there are the only stationary distributions for that Markov chain.

Question:Suppose that a point (X,Y) is to be chosen from the squareSin thexy-plane containing all points (x,y) such that 0鈮�x鈮�1 and 0鈮�y鈮�1. Suppose that the probability that the chosen point will be the corner(0,0)is 0.1, the probability that it will be the corner(1,0)is 0.2, and the probability that it will be the corner(0,1)is 0.4, and the probability that it will be the corner(1,1)is 0.1. Suppose also that if the chosen point is not one of the four corners of the square, then it will be an interior point of the square and will be chosen according to a constant p.d.f. over the interior of the square. Determine

\(\begin{array}{l}\left( {\bf{a}} \right)\;{\bf{Pr}}\left( {{\bf{X}} \le \frac{{\bf{1}}}{{\bf{4}}}} \right)\;{\bf{and}}\\\left( {\bf{b}} \right)\;{\bf{Pr}}\left( {{\bf{X + Y}} \le {\bf{1}}} \right)\end{array}\)
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