Question

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11. What are the mean and variance of the number in the sample that belong to the ruling clique?

12. What are the answers to (1) and (2) if the club has 50,000 members, 10,000 in the ruling clique and 40,000 second-class members.

13. Huck and Jim are waiting for a raft. The number of rafts floating by over intervals of time is a Poisson process with a rate of lambda = 0:4 rafts per day. They agree in advance to let the first raft go and take the second one that comes along. What is the probability that they will have to wait more than a week? Hint: If they have to wait more than a week, what does that say about the number of rafts in a period of 7 days?

14. Fire ant colonies occur according to a spatial Poisson process with a rate of 1.5 colonies per acre. What is the probability that a 10 acre plot of land will have 10 or fewer fire ant colonies?

15. What are the mean and variance of the number of colonies on a 10 acre plot?

Answer 1

11 and 12

What are the mean and variance of the number in the sample that belong to the ruling clique? Mean = 6(10/50) = 1.2 Variance = 6(10/50) (40/50)(1 – 5/49) = 0.862 What are the answers to (1) and (2) if the club has 50,000 members, 10,000 in the ruling clique and 40,000 second-class members. Here, m = 10000, n = 40000 and k = 6 PX > 3) = 1 - P(X<2)= 1 - phyper(2,10000,40000,6)=0.09887 Mean = 6(10000/50000) = 1.2 Variance = 6(10000/50000)(40000/50000) (1 - 5/49999) = 0.9599

Note: p= .2 if we use binomial, P(X> 3) = 1 - P(X<2)= 1 - pbinom(2,6,0.2) = 0.09888

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