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Refer to Exercise 5.86. Suppose that Z is a standard
Chapter 5, Problem 156SE(choose chapter or problem)
Refer to Exercise 5.86. Suppose that \(Z\) is a standard normal random variable and that \(Y\) is an independent \(\chi^{2}\) random variable with \(v\) degrees of freedom.
a Define \(\chi^{2}\). Find \(\operatorname{Cov}(Z, W)\). What assumption do you need about the value of \(v\) ?
b With \(Z\), \(Y\) and \(W\) as above, find \(\operatorname{Cov}(Y, W)\).
c One of the covariances from parts (a) and (b) is positive, and the other is zero. Explain why.
Equation Transcription:
Text Transcription:
Z
Y
x^2
v
W=Z/ sqrt Y
Cov(Z,W)
v
Z,Y
W
Cov(Y,W)
Questions & Answers
QUESTION:
Refer to Exercise 5.86. Suppose that \(Z\) is a standard normal random variable and that \(Y\) is an independent \(\chi^{2}\) random variable with \(v\) degrees of freedom.
a Define \(\chi^{2}\). Find \(\operatorname{Cov}(Z, W)\). What assumption do you need about the value of \(v\) ?
b With \(Z\), \(Y\) and \(W\) as above, find \(\operatorname{Cov}(Y, W)\).
c One of the covariances from parts (a) and (b) is positive, and the other is zero. Explain why.
Equation Transcription:
Text Transcription:
Z
Y
x^2
v
W=Z/ sqrt Y
Cov(Z,W)
v
Z,Y
W
Cov(Y,W)
ANSWER:
Solution:
Step 1 of 3:
Let Z is a standard normal random variable and that Y is an independent random variable with v degrees of freedom
- The claim is to and find Cov(Z, W). and what assumption do we need about the value of v.
define W =
Since, E(Z) = E(W) = 0
Cov(Z, W) = E(ZW)
= E()
= E() E(
)
= E()
Let,
After applying integration by parts then we get:
- (1)
We have, v > -2
Then,
Substitute in equation (1)
Therefore,
Then, Cov(Z, W) = E()
=
Hence, Cov(Z, W) = , v>1
V is greater than one.