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Statistics / Introduction to Probability 1 / Chapter 7 / Problem 66

Introduction to Probability | 1st Edition | ISBN: 9781466575578 | Authors: Joseph K. Blitzstein, Jessica Hwang

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A certain course has a freshmen, b sophomores, c juniors, and d seniors. Let X be the

Chapter 7, Problem 66

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QUESTION:

A certain course has a freshmen, b sophomores, c juniors, and d seniors. Let X be the number of freshmen and sophomores (total), Y be the number of juniors, and Z be the number of seniors in a random sample of size n, where for Part (a) the sampling is with replacement and for Part (b) the sampling is without replacement (for both parts, at each stage the allowed choices have equal probabilities). (a) Find the joint PMF of X, Y, Z, for sampling with replacement. (b) Find the joint PMF of X, Y, Z, for sampling without replacement.

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QUESTION:

A certain course has a freshmen, b sophomores, c juniors, and d seniors. Let X be the number of freshmen and sophomores (total), Y be the number of juniors, and Z be the number of seniors in a random sample of size n, where for Part (a) the sampling is with replacement and for Part (b) the sampling is without replacement (for both parts, at each stage the allowed choices have equal probabilities). (a) Find the joint PMF of X, Y, Z, for sampling with replacement. (b) Find the joint PMF of X, Y, Z, for sampling without replacement.

ANSWER:

Step 1 of 2

.

If the students are selected with replacement, then the marginal and conditional distributions will be multinomial. Therefore, we have

Thus, the joint PMF of for sampling with replacement is:

.

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