sol4S09 - PSTAT 120C Solutions for Assignment#4 1 Suppose...

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PSTAT 120C: Solutions for Assignment #4 May 19, 2009 1. Suppose that we know that of the large fish in a lake the species are 40% perch, 35% bass and 25% trout. (a) The probability that a random sample of 5 large fish contains 2 perch, 2 bass, and 1 trout is P { X p = 2 , X b = 2 , X t = 1 } = 5 2 2 1 (0 . 4) 2 (0 . 35) 2 (0 . 25) = 5! 2!2! (0 . 16)(0 . 1225)( . 25) = 0 . 147 (b) A random sample of 4 large fish will have more trouts than bass if there are 3 or 4 trout or if there are 2 trout and either 1 or 0 bass and if there is 1 trout, 1 bass and 2 perch. P { 4 trout } = (0 . 25) 4 = 0 . 00391 P { 3 trout } = 4 3 (0 . 25) 3 (0 . 75) = 0 . 046875 P { 2 trout and 1 bass } = 4! 2!1!1! (0 . 25) 2 (0 . 35)( . 4) = 0 . 105 P { 2 trout and no bass } = 4! 2!0!2! (0 . 25) 2 ( . 4) 2 = 0 . 06 P { 1 trout and no bass } = 4! 1!0!3! (0 . 25)( . 4) 3 = 0 . 064 so that the total probability is 0.2798. 2. Suppose that X 1 , X 2 , X 3 , X 4 are multinomially distributed with n = 16 and p 1 = ab p 2 = (1 - a ) b (1) p 3 = a (1 - b ) p 4 = (1 - a )(1 - b ) (2) for some a and b between 0 and 1. Let S = X 1 + X 2 and R = X 1 + X 3 . (a) In lecture, we argued that if we add to of the observations from a multinomial together the result is still a multinomial. Therefore,
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