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55. Given this joint pmf, (a) Find the conditional distribution of X given Y = 0, fx|y=0(x) = P(X = x|Y = 0). (b) Are X and Y independent? Justify your answer.

55. Given this joint pmf, (a) Find the conditional distribution of X given Y = 0, fx|y=0(x) = P(X = x|Y = 0). (b) Are X and Y independent? Justify your answer.

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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A small department has 10 faculty members of whom 3 are assistant professors (aP), 5 are associate professors (AP), and 2 are professors (P). Of these 10 faculty members, 2 are randomly selected to be on a committee. Let X denote the number of aP selected and let Y denote the number of AP selected. X and Y are jointly discrete random variables with mass points (x, y) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)}. Moreover, we can find the probability associated with each of these mass points as follows: f(0,0)=P(X= 0, Y = 0) = P(2 P selected) 4/55 ƒ(0, 1) = P(X = 0, Y = 1) = P(1 P, 1 AP selected) = 9 = X = Number of aP chosen 0 1 2 1/45 6/45 3/45 10/45 f(1,1) 0 10/45 0 0 (3) = Y = Number of AP chosen 0 1 2 ⠀ = P(X = 2, Y = 0) = P(2 aP selected) == Leading to the joint pmf: f(x, y) = P(X = x, Y = y) = = H
Transcribed Image Text:A small department has 10 faculty members of whom 3 are assistant professors (aP), 5 are associate professors (AP), and 2 are professors (P). Of these 10 faculty members, 2 are randomly selected to be on a committee. Let X denote the number of aP selected and let Y denote the number of AP selected. X and Y are jointly discrete random variables with mass points (x, y) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)}. Moreover, we can find the probability associated with each of these mass points as follows: f(0,0)=P(X= 0, Y = 0) = P(2 P selected) 4/55 ƒ(0, 1) = P(X = 0, Y = 1) = P(1 P, 1 AP selected) = 9 = X = Number of aP chosen 0 1 2 1/45 6/45 3/45 10/45 f(1,1) 0 10/45 0 0 (3) = Y = Number of AP chosen 0 1 2 ⠀ = P(X = 2, Y = 0) = P(2 aP selected) == Leading to the joint pmf: f(x, y) = P(X = x, Y = y) = = H
55. Given this joint pmf, (a) Find the conditional distribution of X given Y = 0, fxY=0(x) = P(X=x|Y = 0). (b) Are X and Y independent? Justify your answer.
Transcribed Image Text:55. Given this joint pmf, (a) Find the conditional distribution of X given Y = 0, fxY=0(x) = P(X=x|Y = 0). (b) Are X and Y independent? Justify your answer.
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