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Suppose that Y is a random variable with moment-generating
Chapter 3, Problem 156E(choose chapter or problem)
Suppose that \(Y\) is a random variable with moment-generating function \(m(t)\).
a What is \(m(0)\)?
b If \(W=3Y\), show that the moment-generating function of \(W\) is \(m(3t)\).
c If \(X=Y?2\), show that the moment-generating function of \(X\) is \(e^{-2 t} m(t)\).
Questions & Answers
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QUESTION:
Suppose that \(Y\) is a random variable with moment-generating function \(m(t)\).
a What is \(m(0)\)?
b If \(W=3Y\), show that the moment-generating function of \(W\) is \(m(3t)\).
c If \(X=Y?2\), show that the moment-generating function of \(X\) is \(e^{-2 t} m(t)\).
ANSWER:Step 1 of 3:
(a) Assume that \(Y\) is a random variable with moment-generating function \(m(t)\)
We need to find the value of \(m(0)\)
The moment-generating function \(m_{Y}(t)\) for a random variable \(Y\) is defined to be,
\(m_{Y}(t)=E\left(e^{t Y}\right)\)
(1)
We are asked to find \(m(0)\), hence put \(t=0\), into the equation (1),
\(\begin{array}{l} m_{Y}(0)=E\left(e^{0 \times Y}\right) \\ m_{Y}(0)=E\left(e^{0}\right)=E(1)=1 \end{array}\)
[since \(E(C)=C, \text { where } C \text { is a constant }\)]
Hence the value of \(m(0)\) is \(1\)
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