Solution Found!
Suppose that Z has a standard normal distribution and that
Chapter 7, Problem 30E(choose chapter or problem)
Suppose that Z has a standard normal distribution and that Y is an independent \(x^{2}\) -distributed random variable with ν df. Then, according to Definition 7.2,
\(\mathrm{T}=\frac{Z}{\sqrt{Y / v}}\)
has a t distribution with \(\mathrm{V} \mathrm{df} .{ }^{1}\)
If Z has a standard normal distribution, give \(E(Z)\) and \(E\left(Z^{2}\right)\). [Hint: For any random variable, \(\mathrm{E}\left(Z^{2}\right)=V(Z)+(E(Z))^{2}\).]Exercises preceded by an asterisk are optional.
According to the result derived in Exercise 4.112(a), if Y has a \(x^{2}\) distribution with ν df, then
\(E\left(Y^{a}\right)=\frac{\Gamma([v / 2]+a)}{\Gamma(v / 2)} 2^{a}, \quad \text { if } v>-2 a\)
Use this result, the result from part (a), and the structure of T to show the following. [Hint: Recall the independence of Z and Y .]
I. \(E(T)=0, \text { if } v>1\)
II. \(V(T)=v /(v-2), \text { if } v>2\)
Equation Transcription:
Text Transcription:
X2
T=Z\sqrt Y / v
v df1
E(Z)
E(Z2)
E(Z2)=V(Z)+(E(Z))2
X2
E Y^a=\\Gamma([v / 2]+a)\Gamma(v / 2) 2^a, \quad if v>-2 a
E(T)=0, if v >1
V(T)=v/(v-2), if v>2
Questions & Answers
QUESTION:
Suppose that Z has a standard normal distribution and that Y is an independent \(x^{2}\) -distributed random variable with ν df. Then, according to Definition 7.2,
\(\mathrm{T}=\frac{Z}{\sqrt{Y / v}}\)
has a t distribution with \(\mathrm{V} \mathrm{df} .{ }^{1}\)
If Z has a standard normal distribution, give \(E(Z)\) and \(E\left(Z^{2}\right)\). [Hint: For any random variable, \(\mathrm{E}\left(Z^{2}\right)=V(Z)+(E(Z))^{2}\).]Exercises preceded by an asterisk are optional.
According to the result derived in Exercise 4.112(a), if Y has a \(x^{2}\) distribution with ν df, then
\(E\left(Y^{a}\right)=\frac{\Gamma([v / 2]+a)}{\Gamma(v / 2)} 2^{a}, \quad \text { if } v>-2 a\)
Use this result, the result from part (a), and the structure of T to show the following. [Hint: Recall the independence of Z and Y .]
I. \(E(T)=0, \text { if } v>1\)
II. \(V(T)=v /(v-2), \text { if } v>2\)
Equation Transcription:
Text Transcription:
X2
T=Z\sqrt Y / v
v df1
E(Z)
E(Z2)
E(Z2)=V(Z)+(E(Z))2
X2
E Y^a=\\Gamma([v / 2]+a)\Gamma(v / 2) 2^a, \quad if v>-2 a
E(T)=0, if v >1
V(T)=v/(v-2), if v>2
ANSWER:
Step 1 of 4
Given:
Z follows a standard normal distribution and Y follows a 
A variable T is defined as follows:
