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Identify the distributions of the random variables with
Chapter 4, Problem 140E(choose chapter or problem)
Identify the distributions of the random variables with the following moment-generating functions:
a \(m(t)=(1-4 t)^{-2}\).
b \(m(t)=1 /(1-3.2 t)\).
c \(m(t)=e^{-5 t+6 t^{2}}\).
Questions & Answers
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QUESTION:
Identify the distributions of the random variables with the following moment-generating functions:
a \(m(t)=(1-4 t)^{-2}\).
b \(m(t)=1 /(1-3.2 t)\).
c \(m(t)=e^{-5 t+6 t^{2}}\).
ANSWER:Step 1 of 3
The objective is to identify the distribution of the random variables for the given moment-generating function.
(a)
From the given information,
The moment-generating function is,
\(m\left( t \right) = {\left( {1 - 4t} \right)^{ - 2}}\)
Let Y have gamma distribution with \(\alpha\) and \(\beta\) then the moment generating function of Y is,
\({M_x}\left( t \right) = {\left( {1 - \beta t} \right)^{ - \alpha }},t < \frac{1}{\beta }\)
Compare the above two moments generating function, then Y has a gamma distribution with \(\alpha = 2\) and \(\beta = 4\).
Therefore, the given moment-generating function is a gamma distribution.
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