(2) Compute EW, .
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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Please do part 2
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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