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April 9th, 2018, 07:50 PM   #1
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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
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April 10th, 2018, 12:38 AM   #2
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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 b-d 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.
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Last edited by romsek; April 10th, 2018 at 01:27 AM.
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April 10th, 2018, 11:41 AM   #3
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Multinomial

Can you, however, help me with the part C and D?


Quote:
Originally Posted by romsek View Post
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 b-d 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.
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April 10th, 2018, 12:19 PM   #4
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Quote:
Originally Posted by Shima View Post
Can you, however, help me with the part C and D?
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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