Find the variance of Y a) We ask someone to taste 12 cookies ( 8 are made with b

Question

Find the variance of Y

a) We ask someone to taste 12 cookies ( 8 are made with butter and 4 with margarine ) and to identify the cookies with butter. He doesnt know the exact number of butter cookie and choose randomly.
Y= number of butter cookies correctly identified

b) We ask someone to taste 12 cookies ( 8 are made with butter and 4 with margarine ) and to identify the cookies with butter. He know the exact number of butter cookie and choose randomly 8 cookies.
Y= number of butter cookies correctly identified

c) 70% of the people in a shopping mall between 9h00 and 18h00 are female and 1 person enter on average every 4 minute.
Y= number of female who will enter the mall in the next 12 minutes

d) 70% of the people in a shopping mall between 9h00 and 18h00 are female and 1 person enter on average every 4 minute.
Y= number of male who will enter the mall in the next 24 person

Explanation / Answer

a) If he doesn't know the exact number of butter cookie then it would be a binomial distribution with:

n = 12, p = 0.5, q = 1 - 0.5 = 0.5

So,

Variance = npq = 12(0.5)(0.5) = 3

b) When he know the exact number of butter cookie and he chooses randomly 8 cookies then it will be a hypergeometric distribution with:

N = 12

n = 8

p = 8/12 = 2/3

q = 1 - 2/3 = 1/3

So,

Variance = npq*(N - n)/(N - 1) = 8(2/3)(1/3)*(12 - 8)/ (12 - 1) = 64/99 = 0.6465

c) This will be a poisson distribution. 1 person enters every 4 minute so in next 12 minutes, average 3 persons will enter.

In that,

Average no of female = 0.7(3) = 2.1

For Poisson distribution,

Mean = Variance

So,

Variance = 2.1

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