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According to an article in the August 30, 2002 issue of the Chronicle of Higher Education, 30% of first-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road. Choose two students at random, let X be the number of liber- als, and let Y be the number of conservatives. a. Using the multinomial distribution from Section 5.1, give the joint probability mass func- tion p(x, y) of X and Y. Give the joint probability table showing all nine values, of which three should be 0. b. Determine the marginal probability mass func- tions by summing pix, y) numerically. How could these be obtained directly? [Hint: What are the univariate distributions of X and Y?] c. Determine the conditional probability mass function of Y given X = x for x = 0, 1, 2. Compare with the Bin[2-x, .2/(.2 + .5)] distri- bution. Why should this work?

According to an article in the August 30, 2002 issue of the Chronicle of Higher Education, 30% of first-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road. Choose two students at random, let X be the number of liber- als, and let Y be the number of conservatives. a. Using the multinomial distribution from Section 5.1, give the joint probability mass func- tion p(x, y) of X and Y. Give the joint probability table showing all nine values, of which three should be 0. b. Determine the marginal probability mass func- tions by summing pix, y) numerically. How could these be obtained directly? [Hint: What are the univariate distributions of X and Y?] c. Determine the conditional probability mass function of Y given X = x for x = 0, 1, 2. Compare with the Bin[2-x, .2/(.2 + .5)] distri- bution. Why should this work?

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 4ECP: Show that the probability of drawing a club at random from a standard deck of 52 playing cards is...
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According to an article in the August 30, 2002 issue of the Chronicle of Higher Education, 30% of first-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road. Choose two students at random, let X be the number of liber- als, and let Y be the number of conservatives. a. Using the multinomial distribution from Section 5.1, give the joint probability mass func- tion p(x, y) of X and Y. Give the joint probability table showing all nine values, of which three should be 0. b. Determine the marginal probability mass func- tions by summing p(x, y) numerically. How could these be obtained directly? [Hint: What are the univariate distributions of X and Y?] c. Determine the conditional probability mass function of Y given X = x for x = 0, 1, 2. Compare with the Bin[2-x, .2/(.2 + 5)] distri- bution. Why should this work? d. Are X and Y independent? Explain. e. Find E(YIX = x) forx = 0, 1, 2. Do this numer- ically and then compare with the use of the formula for the binomial mean, using the bino- mial distribution given in part (c). Is E(YX = x) a linear function of x? f. Determine V(YIX = x) for x = 0, 1, 2. Do this numerically and then compare with the use of the formula for the binomial variance, using the binomial distribution given in part (c).
Transcribed Image Text:According to an article in the August 30, 2002 issue of the Chronicle of Higher Education, 30% of first-year college students are liberals, 20% are conservatives, and 50% characterize themselves as middle-of-the-road. Choose two students at random, let X be the number of liber- als, and let Y be the number of conservatives. a. Using the multinomial distribution from Section 5.1, give the joint probability mass func- tion p(x, y) of X and Y. Give the joint probability table showing all nine values, of which three should be 0. b. Determine the marginal probability mass func- tions by summing p(x, y) numerically. How could these be obtained directly? [Hint: What are the univariate distributions of X and Y?] c. Determine the conditional probability mass function of Y given X = x for x = 0, 1, 2. Compare with the Bin[2-x, .2/(.2 + 5)] distri- bution. Why should this work? d. Are X and Y independent? Explain. e. Find E(YIX = x) forx = 0, 1, 2. Do this numer- ically and then compare with the use of the formula for the binomial mean, using the bino- mial distribution given in part (c). Is E(YX = x) a linear function of x? f. Determine V(YIX = x) for x = 0, 1, 2. Do this numerically and then compare with the use of the formula for the binomial variance, using the binomial distribution given in part (c).
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