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(2) Compute EW, .

(2) Compute EW, .

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Problem 3 i.e. Let W, be a negative-binomial variable with parameter p € (0,1), P{W, = k} = p' (1-p)-rifk = r,r+1,..., else. Let furthermore (ilieN be a sequence of independent Bernoulli variables with parameter p. Re- call that W, has the same distribution as Y₁ = k X, := min {k € N₁: [{i=r}, i=1 i.e. the number of independent Bernoulli trials with parameter p that we have to observe until we see the rth success. (1) Define the random variables Yo, Y₁,..., Y, by 0 min {k EN: EY+-+Y₁-1+k=1} for i = 0, for i = 1,...,r. (a) Compute P{Y; = k | Y₁ = ₁,...,Y₁-1=li-1} for positive integers k, l1,...,li-1. (b) It is possible to show that Y₁, Y2,... are independent. Give a heuristic explanation for this fact. (c) Show by induction over rEN, that X, Y₁ +...+Yr. = (2) Compute EW,.
Transcribed Image Text:Problem 3 i.e. Let W, be a negative-binomial variable with parameter p € (0,1), P{W, = k} = p' (1-p)-rifk = r,r+1,..., else. Let furthermore (ilieN be a sequence of independent Bernoulli variables with parameter p. Re- call that W, has the same distribution as Y₁ = k X, := min {k € N₁: [{i=r}, i=1 i.e. the number of independent Bernoulli trials with parameter p that we have to observe until we see the rth success. (1) Define the random variables Yo, Y₁,..., Y, by 0 min {k EN: EY+-+Y₁-1+k=1} for i = 0, for i = 1,...,r. (a) Compute P{Y; = k | Y₁ = ₁,...,Y₁-1=li-1} for positive integers k, l1,...,li-1. (b) It is possible to show that Y₁, Y2,... are independent. Give a heuristic explanation for this fact. (c) Show by induction over rEN, that X, Y₁ +...+Yr. = (2) Compute EW,.
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