Solution Found!
Answer: Suppose that Y is a discrete random variable with
Chapter 3, Problem 32E(choose chapter or problem)
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) = ,