Solution Found!
Let Z be a standard normal random variable and let Y1 = Z
Chapter 5, Problem 100E(choose chapter or problem)
Let \(Z\) be a standard normal random variable and let \(Y_{1}=Z and Y_{2}=Z^{2}\).
a What are \(E\left(Y_{1}\right) and E\left(Y_{1}\right)\)?
b What is \(E\left(Y_{1} Y_{2}\right)\) [Hint: \(\mathbf{E}\left(Y_{1} Y_{2}\right)=E\left(Z^{3}\right)\), recall Exercise 4.199.]
c What is \({Cov}\left(Y_{1}, Y_{2}\right)\)?
d Notice that \(P\left(Y_{2}>1 \mid Y_{1}>1\right)=1\).Are \(Y_{1}\) and \(Y_{2}\) independent?
Equation Transcription:
Text Transcription:
Z
Y_1=Z
Y_2=Z^2
E(Y_1)
E(Y_2)
E(Y1Y_2)=E(Z^3)
Cov(Y_1,Y_2)
P(Y_2>1|Y_1>1=1
Y_1
Y_2
Questions & Answers
QUESTION:
Let \(Z\) be a standard normal random variable and let \(Y_{1}=Z and Y_{2}=Z^{2}\).
a What are \(E\left(Y_{1}\right) and E\left(Y_{1}\right)\)?
b What is \(E\left(Y_{1} Y_{2}\right)\) [Hint: \(\mathbf{E}\left(Y_{1} Y_{2}\right)=E\left(Z^{3}\right)\), recall Exercise 4.199.]
c What is \({Cov}\left(Y_{1}, Y_{2}\right)\)?
d Notice that \(P\left(Y_{2}>1 \mid Y_{1}>1\right)=1\).Are \(Y_{1}\) and \(Y_{2}\) independent?
Equation Transcription:
Text Transcription:
Z
Y_1=Z
Y_2=Z^2
E(Y_1)
E(Y_2)
E(Y1Y_2)=E(Z^3)
Cov(Y_1,Y_2)
P(Y_2>1|Y_1>1=1
Y_1
Y_2
ANSWER:
Solution 100E
Step1 of 5:
Let us consider a standard normal random variable and let .
Here our goal is:
a). We need to find
b). We need to find
c). We need to find
d). We need to check whether are independent or not by noticing
Step2 of 5:
a).
We know that the standard normal random variable is follows normal distribution with parameter where,
That is
Standard normal density function is given by:
Now,
We know that for an odd function
Therefore,
Similarly,