Quiz 11 Answers.pdf - 1 Suppose we flip a coin 10 times and...

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1. Suppose we flip a coin 10 times and we record 8 heads, 2 tails. Let p be the true probability of flipping heads. (a) Given this data, a likelihood function L for the parameter p . L ( p ) = Pr(8 heads | p ) = 10 8 p 8 (1 - p ) 2 = 45 p 8 (1 - p ) 2 . (b) Find all the critical points of L . L 0 ( p ) = 45(8 p 7 (1 - p ) 2 - 2 p 8 (1 - p )) = 45(1 - p ) p 7 (8(1 - p ) - 2 p ) . So L 0 ( p ) = 0 when p = 0, p = 1 or 0 = 8(1 - p ) - 2 p = 8 - 8 p - 2 p = 8 - 10 p = p = 8 / 10 = . 8 . (c) Find the maximum likelihood estimator ˆ p . L (0) = 0 and L (1) = 0 but L ( . 8) = 45( . 8) 8 ( . 2) 2 0 . 302. Since 0 . 302 is greater than 0, ˆ p = 0 . 302. 2. A computer network experiences 4 disconnections in one minute, 2 in another minute and 7 in another minute. Assume the number N of disconnections in one minute follows a Poisson distribution with parameter Λ, so Pr( N = k ) = e - Λ Λ k k ! . (a) Given this data, a likelihood function L for the parameter Λ. We are given t = 1, so L (Λ) = Pr( N = 4) Pr( N = 2) Pr( N = 7) = e - Λ Λ 4 4! · e - Λ Λ 2 2! · e - Λ Λ 7 7! = e - Λ 13 4!2!7! (b) Find all the critical points of L . L 0 (Λ) = - 3 e - Λ 13 + 13 e - Λ 12 4!2!7! = e - Λ 12 ( - 3Λ + 13) 4!2!7! So L 0 (Λ) = 0 only when Λ = 0 or Λ = 13 / 3. (c) Find the maximum likelihood estimator Λ. Checking, L (0) = 0 but L 13 3 = e - 13 13 13 3 13 4!2!7! 0 . 0018 . Since 0.0018 is greater than 0, ˆ Λ = 13
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  • Spring '17
  • Edward S. Letzter
  • Normal Distribution, Estimation theory, parameter λ, Λ Λ4 e

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