Sol_hw_3 - ASSIGNMENT 3 SOLUTIONS 1 Exercise 1 The rst...

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Unformatted text preview: ASSIGNMENT 3 - SOLUTIONS 1. Exercise 1 The rst algorithm of simulation is the inverse method, applied to this distribution. The pseudo-code is the following: (1) α := ∑ k j =0 e- λ λ j j ! (2) p := α- 1 e- λ i := 0 F := p (3) U := Uniform (0 , 1) (4) while ( U > F ) i := i + 1 p := p λ i F := F + p (5) Return F The second method to generate the random variable is applying the acceptance/rejection method. We will use a discrete uniformly distributed random variable over the set { ,...,k } as auxiliary distribution. Let α = ∑ k j =0 e- λ λ j /j ! . Since we know that the mode of the Poisson distribution is d λ e , we have that e- λ λ j /j ! ≤ e- λ λ d λ e / d λ e ! ∀ j . Our bound for this case will be C = α- 1 e- λ λ d λ e / d λ e ! . De ne the desired distribution probability mass function as p ( i ) = P { X = i } and q ( i ) = 1 / ( k + 1) i = 0 ...k (note that our bound can be written as C = p ( d λ e ) ). The method is then de ned as follows: (1) α := ∑...
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  • Spring '11
  • JoseBlanchet
  • Probability theory, Discrete probability distribution, Probability mass function, Inverse transform sampling, Örm

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