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Statistics / Mathematical Statistics with Applications 7 / Chapter 6 / Problem 37E

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

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Let Y1, Y2, . . . , Yn be independent and identically

Chapter 6, Problem 37E

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QUESTION:

Problem 37E

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(Yi = 0) = q = 1 − p. (Such random variables are called Bernoulli random variables.)

a Find the moment-generating function for the Bernoulli random variable Y1.

b Find the moment-generating function for W = Y1 + Y2 + ··· + Yn .

c What is the distribution of W ?

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QUESTION:

Problem 37E

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(Yi = 0) = q = 1 − p. (Such random variables are called Bernoulli random variables.)

a Find the moment-generating function for the Bernoulli random variable Y1.

b Find the moment-generating function for W = Y1 + Y2 + ··· + Yn .

c What is the distribution of W ?

ANSWER:

Solution:

Step 1 of 3:

Let ${Y}_{1}$ , ${Y}_{2},$ …, ${Y}_{n}$ be independent and identically distributed random variables such that for 0<p<1, P( ${Y}_{i}=1)=p$ and P( ${Y}_{i}=0)=q.$ $\$

We have to find,

The moment-generating function for the bernoulli random variable Y1.
The moment- generating function for W= ${Y}_{1}$ + ${Y}_{2}$ +...+ ${Y}_{n}$ .
The distribution of W.

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