Concept explainers
Refer to Exercises 4.141 and 4.137. Suppose that Y is uniformly distributed on the interval (0, 1) and that a > 0 is a constant.
- a Give the moment-generating
function for Y. - b Derive the moment-generating function of W = aY. What is the distribution of W? Why?
- c Derive the moment-generating function of X = −aY. What is the distribution of X? Why?
- d If b is a fixed constant, derive the moment-generating function of V = aY + b. What is the distribution of V? Why?
a.
Find the moment-generating function for Y.
Answer to Problem 142E
The moment-generating function of Y is
Explanation of Solution
Let Y be a random variable that has a uniform distribution on the interval
The probability density function of Y is given by
The moment-generating function of Y is derived below:
Thus, the moment-generating function of Y is
b.
Find the moment-generating function of
Find the distribution of W.
Answer to Problem 142E
The moment-generating function of W is
The distribution of W is uniform distribution on the interval
Explanation of Solution
The moment-generating function of
Thus, the moment-generating function of W is
Hence, by the uniqueness theorem of mgf, the distribution of W is a uniform distribution on the interval
c.
Derive the moment-generating function of
Find the distribution of X.
Answer to Problem 142E
The moment-generating function of X is
The variable X is a uniform distribution on the interval
Explanation of Solution
The moment-generating function of
The moment- generating function of X is
This indicates that the variable X is a uniform distribution on the interval
d.
Derive the moment-generating function of
Find the distribution of V.
Answer to Problem 142E
The moment-generating function of V is
The variable V is a uniform distribution on the interval
Explanation of Solution
The moment-generating function of
The moment-generating function of V is
This indicates that the variable V is a uniform distribution on the interval
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Chapter 4 Solutions
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