Solution Found!
Refer to Exercises 3.147 and 3.158. If Y has a geometric
Chapter 3, Problem 161E(choose chapter or problem)
Refer to Exercises and 3.158. If
has a geometric distribution with success probability
, consider \(Y^{\star}=Y-1\). Show that the moment-generating function of \(Y^{\star}\) is \(m^{\star}(t)=\frac{p}{1-q e^{t}}\), where \(q=1-p\).
Equation Transcription:
Text Transcription:
Y^star=Y-1
Y^star
m^star (t)=p over 1-qe^t
q=1-p
Questions & Answers
QUESTION:
Refer to Exercises and 3.158. If
has a geometric distribution with success probability
, consider \(Y^{\star}=Y-1\). Show that the moment-generating function of \(Y^{\star}\) is \(m^{\star}(t)=\frac{p}{1-q e^{t}}\), where \(q=1-p\).
Equation Transcription:
Text Transcription:
Y^star=Y-1
Y^star
m^star (t)=p over 1-qe^t
q=1-p
ANSWER:
Solution :
Step 1 of 1:
Given Y has a geometrical distribution with probability p.
Then the moment-generating function Y is
m(t)=
Our goal is:
We need to prove that the moment generating function is