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The conditional expectation E[X|Y] = «(Y), where (y) E(X |Y = y), is a random variable (function of Y) and if EX| < 0, then EX = E[E(X |Y)]. (a). Let Z and W be two discrete rvs, taking values z = 0, 1, 2. and w 0, 1, 2, ., respectively. Using the result given above, show that %3D .... 00 P(Z = k) EP(Z = k | W = j)P(W = j), %3D j-0 EZ = EE(Z|W = j)P(W = j).

The conditional expectation E[X|Y] = «(Y), where (y) E(X |Y = y), is a random variable (function of Y) and if EX| < 0, then EX = E[E(X |Y)]. (a). Let Z and W be two discrete rvs, taking values z = 0, 1, 2. and w 0, 1, 2, ., respectively. Using the result given above, show that %3D .... 00 P(Z = k) EP(Z = k | W = j)P(W = j), %3D j-0 EZ = EE(Z|W = j)P(W = j).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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2. The conditional expectation E[X|Y] = v(Y), where (y):= E(X | Y = y), is a random variable (function of Y) and if EX| < o0, then EX = E[E(X |Y)]. %3D (a). Let Z andW be two discrete rvs, taking values z = 0, 1, 2... and w = 0, 1, 2, ., respectively. Using the result given above, show that %3D P(Z = k) EP(Z = k | W = j)P(W = j), %3D j-0 EZ EE(Z|W = j)P(W = j). %3D j-0 (b). Let Z and W be two independent continuous random variables with density f(1) and p(r), respectively. Using the result given above, calculate P(Z <W) and find the distribution of Z+ W. (c). Let (Z, W) be a continuous random vector with joint density f(2, w). Calculate P(Z <W) and find the distribution of Z + W.
Transcribed Image Text:2. The conditional expectation E[X|Y] = v(Y), where (y):= E(X | Y = y), is a random variable (function of Y) and if EX| < o0, then EX = E[E(X |Y)]. %3D (a). Let Z andW be two discrete rvs, taking values z = 0, 1, 2... and w = 0, 1, 2, ., respectively. Using the result given above, show that %3D P(Z = k) EP(Z = k | W = j)P(W = j), %3D j-0 EZ EE(Z|W = j)P(W = j). %3D j-0 (b). Let Z and W be two independent continuous random variables with density f(1) and p(r), respectively. Using the result given above, calculate P(Z <W) and find the distribution of Z+ W. (c). Let (Z, W) be a continuous random vector with joint density f(2, w). Calculate P(Z <W) and find the distribution of Z + W.
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