Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: (x – (+o²t))² , o?4 + ut + xt 25 2g2
Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: (x – (+o²t))² , o?4 + ut + xt 25 2g2
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Exercise 6. Using the MGF given by Equation (24), deduce the expressions
for the mean and variance we found earlier.
Exercise 7. Prove the algebraic identity:
2g2
+ xt =
+ ut
25
2g2
Transcribed Image Text:Example: The Gamma Distribution
Suppose X ~ T(a, B), then
eetz
dx
BaT(a)
M(t) =
(22)
Now we do a little high school algebra. Define & =
B
Par. Then one may write
1-Bt
Equation (22) as:
M (t) = ()" J, 5aT(@)
(23)
dx
One recognizes the integrand of Equation (23) as a r(a, 6) density, hence
1
M(1) = (r"() =
(24)
%3D
(1 – Bt)a
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