Question

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Question 1. (Multinomial and multivariate hypergeometric distribution. See Lecture 26.) In a mixed MathStats class of 30 students, 20 are enrolled only in statistics and 5 are enrolled only in mathematics: there are 5 students that are enrolled in both statistics and mathematics.

(a) For each Monday of the 12 weeks of semester, a student from the class is selected at random to give a class presentation. This is sampling with replacement: use the multinomial distribution to find the probability that over the 12 week semester there are (exactly) 6 presentations from mathematics-only students and 4 from statistics-only students.

(b) In week 12, the assignments of five different students from the class are selected at random for a marking audit. What is the probability that the selection includes exactly 1 from a mathematics- only student and exactly 3 from statistics-only students.

Answer 1

a)The pmf of multinomial random variable is defined as

,  where . This is sampling with replacement:

Here probability of selcting mathematics-only students is

probability of selcting statistics-only students is .

probability of selcting mathematics-statistics students is

Here .

The probability that over the 12 week semester there are (exactly) 6 presentations from mathematics-only students and 4 from statistics-only students is

b) The probability that the selection includes exactly 1 from a mathematics- only student and exactly 3 from statistics-only students. This is sampling without replacement: The required probabiliity is

.

Multivariate hypergeometric distribution: This is sampling without replacement: Selecting of the kind. Total selections .