3.71 Let Y denote a geometric random variable with probability of success p. a Show that for a positive integer a, P(Y > a) = qª. b Show that for positive integers a and b, P(Y > a + b|Y > a) = q° = P(Y > b). This result implies that, for example, P(Y > 7|Y > 2) = P(Y > 5). Why do you think this property is called the memoryless property of the geometric distribution? c In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious"?
3.71 Let Y denote a geometric random variable with probability of success p. a Show that for a positive integer a, P(Y > a) = qª. b Show that for positive integers a and b, P(Y > a + b|Y > a) = q° = P(Y > b). This result implies that, for example, P(Y > 7|Y > 2) = P(Y > 5). Why do you think this property is called the memoryless property of the geometric distribution? c In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious"?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:3.71
Let Y denote a geometric random variable with probability of success p.
a Show that for a positive integer a,
P(Y > a) = qª.
b Show that for positive integers a and b,
P(Y > a + b|Y > a) = q° = P(Y > b).
This result implies that, for example, P(Y > 7|Y > 2) = P(Y > 5). Why do you think
this property is called the memoryless property of the geometric distribution?
c In the development of the distribution of the geometric random variable, we assumed
that the experiment consisted of conducting identical and independent trials until the first
success was observed. In light of these assumptions, why is the result in part (b) “obvious"?
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