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Unformatted text preview: Final Practice Problems  Answer Key 1. The following is a fictional joint probability distribution for the age of an individual and the neigh borhood of San Diego in which they live. Neighborhood/Age 25 years 35 years 60 years Pacific Beach 0.2 0.1 North Park 0.1 0.2 0.1 La Jolla 0.1 0.2 (a) What is the probability of being 25 years old? What is the probability of being 35 years old? What is the probability of being 60 years old? Let A denote age and N neighborhood. To find the marginal probability just note that P ( A = 25) = P ( A = 25 ,N = PB ) + P ( A = 25 ,N = NP ) + P ( A = 25 ,N = LJ ) = 0 . 3 . Similarly we can obtain P ( A = 35) = 0 . 4 and P ( A = 60) = 0 . 3. (b) What is the probability of living in La Jolla conditional on being 60 years old? Using the formula for conditional probability, P ( N = LJ  A = 60) = P ( N = LJ,A = 60) /P ( A = 60) = 0 . 2 / . 3 = 2 / 3 . (c) What is the probability of living in Pacific Beach conditional on being 60 years old? As in part (b), P ( N = PB  A = 60) = P ( N = PB,A = 60) /P ( A = 60) = 0 / . 3 = 0. (d) What is the mean of age? What is the variance of age? Age is a discrete random variable, for which we have found the pmf in part (a). Hence, E [ A ] = 25 . 3 + 35 . 4 + 60 . 3 = 39 . 5 . Similarly, using the formula for variance of a discrete random variable we get: V ar ( A ) = (25 39 . 5) 2 . 3 + (35 39 . 5) 2 . 4 + (60 39 . 5) 2 . 3 = 197 . 25 . (e) Are age and neighborhood where people live independent? Justify your answer. Since P ( N = LJ ) = 0 . 3, we obtain that: P ( A = 25 ,N = LJ ) = 0 6 = 0 . 3 . 3 = P ( A = 25) P ( N = LJ ) , and therefore neighborhood and age are not independent. 2. Suppose each observation X i has a normal distribution with unknown mean and variance 2 = 1. Recall this implies that if you sample N observations then X has exactly a normal distribution with mean and variance 2 /N . Suppose you sample 25 observations and find that X = 12....
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 Spring '10
 Elliot

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