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The moment-generating function for the gamma random
Chapter 4, Problem 143E(choose chapter or problem)
The moment-generating function for the gamma random variable is derived in Example 4.13.
Differentiate this moment-generating function to find the mean and variance of the gamma distribution.
Find the moment-generating function for a gamma-distributed random variable.
\(m(t)=E\left(e^{ty}\right)=\int_0^{\infty}e^{ty}\left[\frac{y^{\alpha-1}e^{-y/\beta}}{\beta^{\alpha}\Gamma(\alpha)}\right]\ dy\)
\(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[-y\left(\frac{1}{\beta}-t\right)\right]\ dy\)
\(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[\frac{-y}{\beta/(1-\beta t)}\right]\ dy\).
Equation Transcription:
Text Transcription:
m(t)=E(e^ty)=integral 0 to infinity e^ty[y^alpha-1 e^-y/beta over beta^alpha Gamma(alpha)]dy
=1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y(1 over beta -t)]dy
=1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y over /(1-beta t)]dy
Questions & Answers
QUESTION:
The moment-generating function for the gamma random variable is derived in Example 4.13.
Differentiate this moment-generating function to find the mean and variance of the gamma distribution.
Find the moment-generating function for a gamma-distributed random variable.
\(m(t)=E\left(e^{ty}\right)=\int_0^{\infty}e^{ty}\left[\frac{y^{\alpha-1}e^{-y/\beta}}{\beta^{\alpha}\Gamma(\alpha)}\right]\ dy\)
\(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[-y\left(\frac{1}{\beta}-t\right)\right]\ dy\)
\(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[\frac{-y}{\beta/(1-\beta t)}\right]\ dy\).
Equation Transcription:
Text Transcription:
m(t)=E(e^ty)=integral 0 to infinity e^ty[y^alpha-1 e^-y/beta over beta^alpha Gamma(alpha)]dy
=1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y(1 over beta -t)]dy
=1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y over /(1-beta t)]dy
ANSWER:
Solution:
Step 1 of 2:
From exercise 4.13.
We have the moment generating function for the gamma distribution is
Here we have to differentiate this mgf and to find the mean and variance.
That is,
(since
If t = 0,
=
= E(Y)
If t = 0,
=