HW04 Answers

# HW04 Answers - Solution for 36217 Wanjie Wang Teacher...

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Solution for 36217 Wanjie Wang Teacher: Jiashun Jin September 23, 2009 If there is any question, please contact me at [email protected] 1.(10 Points) Show that X =3 is the most likely outcome. The PMF for Binomial(6, 1/2) is p X ( k ) = P ( X = k ) = ± 6 k ² (1 / 2) k (1 - 1 / 2) 6 - k = ± 6 k ² 1 64 So, P ( X = 0) = 1 / 64 , P ( X = 1) = 6 / 64 , P ( X = 2) = 15 / 64 , P ( X = 3) = 20 / 64 , P ( X = 4) = 15 / 64 , P ( X = 5) = 6 / 64 , P ( X = 6) = 1 / 64. Obviously the X =3 is the most likely outcome. 2. (10 Points)Show the monotonicity of P ( X = k ) ··· The PMF for Poisson distribution is p X ( k ) = e - λ λ k k ! , k = 0 , 1 , 2 , ··· P ( X = k ) P ( X = k - 1) = e - λ λ k k ! e - λ λ k - 1 ( k - 1)! = λ k So, for k < λ , P ( X = k ) > P ( X = k - 1), which means that P ( X = k ) increases monotonically as k increases. As k increases to k λ , P ( X = k ) P ( X = k - 1), which means that P ( X = k ) decreases monotonically as k decreases. So, P ( X = k ) reaches its maximum when k is the largest integer not exceeding λ . 3. Compute the probability that you will win a prize ··· The distribution of number of prize is Binomial(50, 1/100). (a) (3 Points) at least once, P ( X = 0) = (99 / 100) 50 0 . 6050 P ( X 1) = 1 - P ( X = 0) 0 . 3950 (b) (3 Points) exactly once, P ( X = 1) = ± 50 1 ² (1 / 100)(99 / 100) 49 0 . 3056 1

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(c) (3 Points) at least twice. P
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HW04 Answers - Solution for 36217 Wanjie Wang Teacher...

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