Answer: Suppose that Y is a discrete random variable with

Chapter 3, Problem 32E

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QUESTION:

Suppose that  is a discrete random variable with mean  and variance \(\sigma^{2}\) and let \(U=Y / 10\).
a Do you expect the mean of  to be larger than, smaller than, or equal to \(\mu=E(Y)\)? Why?
b Use Theorem  to express \(E(U)=E(Y / 10)\) in terms of \(\mu=E(Y)\). Does this result agree with your answer to part (a)?

c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of  to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y)\)? Why?
d Use Definition  and the result in part (b) to show that

                   \(V(U)=E\left\{[U-E(U)]^{2}\right\}=E\left[.01(Y-\mu)^{2}\right]=.01 \sigma^{2}\);

that is, \(U=Y / 10\) has variance  times that of .

Equation Transcription:

Text Transcription:

mu

sigma^2

U=Y/10

mu=E(Y)

E(U)=E(Y/10)

mu=E(Y)

sigma^2=V(Y)

V(U)=E{[U-E(U)]^2}=E[.01(Y-mu)^2]=.01sigma^2

U=Y/10

Questions & Answers

QUESTION:

Suppose that  is a discrete random variable with mean  and variance \(\sigma^{2}\) and let \(U=Y / 10\).
a Do you expect the mean of  to be larger than, smaller than, or equal to \(\mu=E(Y)\)? Why?
b Use Theorem  to express \(E(U)=E(Y / 10)\) in terms of \(\mu=E(Y)\). Does this result agree with your answer to part (a)?

c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of  to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y)\)? Why?
d Use Definition  and the result in part (b) to show that

                   \(V(U)=E\left\{[U-E(U)]^{2}\right\}=E\left[.01(Y-\mu)^{2}\right]=.01 \sigma^{2}\);

that is, \(U=Y / 10\) has variance  times that of .

Equation Transcription:

Text Transcription:

mu

sigma^2

U=Y/10

mu=E(Y)

E(U)=E(Y/10)

mu=E(Y)

sigma^2=V(Y)

V(U)=E{[U-E(U)]^2}=E[.01(Y-mu)^2]=.01sigma^2

U=Y/10

ANSWER:

 Solution:

Step 1 of 5:

  Suppose that Y is a discrete random variable with mean  and variance  and

 let .

 E(Y) = ,  

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