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assign4 - PSTAT 120C Assignment#4 Due 1 Suppose that we...

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PSTAT 120C: Assignment #4 Due May 14, 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) What is the probability that a random sample of 5 large fish contains 2 perch, 2 bass, and 1 trout? (b) What is the probability that a random sample of 4 large fish will have more trouts than bass? 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) Find P { R = 10 } . (b) Find the P { X 1 = 6 | S = 12 } . (c) Suppose we want to condition on the event { S = 12 , R = 10 } , what are the possible values that X 1 can take? (d) Find P { X 1 = 0 | S = 12 , R = 10 } . 3. A gambler wants to test whether the die she has is fair so she rolls it 300 times and records the outcomes as Roll 1 2 3 4 5 6 Count 54 71 42 45 51 37 meaning she rolled a 1 on the die 54 times, etc. (a) A fair die has probability 1/6 for each of the six outcomes. Use a chi-squared test to test whether or not the probabilities are all equal.
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