3. Consider throws of a biased coin with probability p of showing a head and q = 1-p of showing a tail. Assume that throws are independent of each other, and let X denote the number of throws until you obtain a head. (c) Now consider the following game. In each round you throw the biased coin until you get a head, which is the end of the round. the number of tails before the head is odd, the game ends. If it is even, you start another round. Let N denote the total number of rounds you play. i. Show that the probability mass function of N is given by PN (n) = Q"-P, n= 1,2, ..., where Q and P have to be specified in terms of p and q. ii. Let Y, denote the total number of throws required to obtain the ith head (after obtaining the (i – 1)th head), i = 1,..., N. Let W be the total number of throws when the game ends. Write W in terms of the Y's.
3. Consider throws of a biased coin with probability p of showing a head and q = 1-p of showing a tail. Assume that throws are independent of each other, and let X denote the number of throws until you obtain a head. (c) Now consider the following game. In each round you throw the biased coin until you get a head, which is the end of the round. the number of tails before the head is odd, the game ends. If it is even, you start another round. Let N denote the total number of rounds you play. i. Show that the probability mass function of N is given by PN (n) = Q"-P, n= 1,2, ..., where Q and P have to be specified in terms of p and q. ii. Let Y, denote the total number of throws required to obtain the ith head (after obtaining the (i – 1)th head), i = 1,..., N. Let W be the total number of throws when the game ends. Write W in terms of the Y's.
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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Need help with i and ii please
Transcribed Image Text:3. Consider throws of a biased coin with probability p of showing a head and q = 1-p
of showing a tail. Assume that throws are independent of each other, and let X
denote the number of throws until you obtain a head.
Transcribed Image Text:(c) Now consider the following game. In each round you throw the biased coin
until you get a head, which is the end of the round.
the number of tails
before the head is odd, the game ends. If it is even, you start another round.
Let N denote the total number of rounds you play.
i. Show that the probability mass function of N is given by
PN (n) = Q"-P, n= 1,2, ...,
where Q and P have to be specified in terms of p and q.
ii. Let Y, denote the total number of throws required to obtain the ith head
(after obtaining the (i – 1)th head), i = 1,..., N.
Let W be the total number of throws when the game ends. Write W in
terms of the Y's.
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