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April 9th, 2018, 08:50 PM  #1 
Newbie Joined: Apr 2018 From: Canada Posts: 3 Thanks: 0  Joint PMF of Multivariate
A class contains 6 freshmen, 15 sophomores, 12 juniors and 7 seniors. Suppose that a random sample of 5 students is taken with replacement. Let x1; x2; x3; and x4 denotes the number of freshmen, sophomores, juniors, and seniors, respectively, obtained. a) Identify and determine the joint pmf ofx1;x2;x3andx4. Use part a) and the FPF to determine the probability the sample contains: b) no freshmen, 3 sophomores, 1 junior and 1 senior. c) at least 3 sophomores. d) at least 3 sophomores and at least one junior 
April 10th, 2018, 01:38 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230 
This is a multinomial distribution. let $p = \dfrac{1}{40}(6,~15,~12,~7)$ $p[x_1,x_2,x_3,x_4] = \dfrac{5!}{\prod \limits_{k=1}^4~x_k!} \cdot \prod \limits_{k=1}^4~(p_k)^{x_k}$ for bd create a list of selections that match the criteria, find the probability of each using the formula above, and sum them up to find the probability of satisfying the criteria. Last edited by romsek; April 10th, 2018 at 02:27 AM. 
April 10th, 2018, 12:41 PM  #3  
Newbie Joined: Apr 2018 From: Canada Posts: 3 Thanks: 0  Multinomial
Can you, however, help me with the part C and D? Quote:
 
April 10th, 2018, 01:19 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230  you can't list the groupings that satisfy the criteria for c and d and plug the numbers in? c will be any selection with 3, 4, or 5 sophmores d will be any selections with 1 junior and 3 or 4 sophmores I have great faith in your ability to list these out and chug the numbers. Use some software. 

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