Suppose X is a gamma random variable gamma(a, B), where a > 1 and B > 0: x > 0 s(2) = { * a-le-/8 otherwise What value of r maximizes the density function of X? Find the moment-generating function of X. Let Y be an independently and identically distributed gamma random variable as X, and let Z = X +Y. Find the moment-generating function of Z. What is the distribution of Z?
Suppose X is a gamma random variable gamma(a, B), where a > 1 and B > 0: x > 0 s(2) = { * a-le-/8 otherwise What value of r maximizes the density function of X? Find the moment-generating function of X. Let Y be an independently and identically distributed gamma random variable as X, and let Z = X +Y. Find the moment-generating function of Z. What is the distribution of Z?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Suppose \( X \) is a gamma random variable \( \text{gamma}(\alpha, \beta) \), where \( \alpha > 1 \) and \( \beta > 0 \):
\[
f(x) =
\begin{cases}
\frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1} e^{-x/\beta} & x > 0 \\
0 & \text{otherwise}
\end{cases}
\]
1. **What value of \( x \) maximizes the density function of \( X \)?**
2. **Find the moment-generating function of \( X \).**
3. Let \( Y \) be an independently and identically distributed gamma random variable as \( X \), and let \( Z = X + Y \). Find the moment-generating function of \( Z \). What is the distribution of \( Z \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbb74936-c73d-423e-b5c0-4a7ff906acfc%2Fa6afa99d-5bae-4dee-9bca-ecf431309383%2F4tiqga_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose \( X \) is a gamma random variable \( \text{gamma}(\alpha, \beta) \), where \( \alpha > 1 \) and \( \beta > 0 \):
\[
f(x) =
\begin{cases}
\frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1} e^{-x/\beta} & x > 0 \\
0 & \text{otherwise}
\end{cases}
\]
1. **What value of \( x \) maximizes the density function of \( X \)?**
2. **Find the moment-generating function of \( X \).**
3. Let \( Y \) be an independently and identically distributed gamma random variable as \( X \), and let \( Z = X + Y \). Find the moment-generating function of \( Z \). What is the distribution of \( Z \)?
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