Solution Found!
Suppose that Z is a standard normal random variable and
Chapter 5, Problem 86E(choose chapter or problem)
Suppose that is a standard normal random variable and that \(Y_{1}\) and \(Y_{2}\) are \(\chi^{2}\)-distributed random variables with \(v_{1}\) and \(v_{2}\) degrees of freedom, respectively. Further, assume that Z, \(Y_{1}\) and \(Y_{1}\) are independent.
a Define \(W=Z / \sqrt{Y_{1}}\). Find \(E(W)\) and \(V(W)\). What assumptions do you need about the value of \(v_{1}\)? [Hint: \(W=Z\left(1 / \sqrt{Y_{1}}\right)=g(Z) h\left(Y_{1}\right)\). Use Theorem 5.9. The results of Exercise 4.112(d) will also be useful.]
b Define \(U=Y_{1} / Y_{2}\). Find \(E(U)\) and \(V(U)\). What assumptions about \(v_{1}\) and \(v_{2}\) do you need? Use the hint from part (a).
Equation Transcription:
Text Transcription:
Y_1
Y_2
chi_2
v_1
v_2
Y_1
Y_2
W=Z/ sqrt Y_1
E(W)
V(W)
v_1
W=Z(1/ sqrt Y_1)=g(Z)h(Y_1)
U=Y_1/Y_2
E(U)
V(U)
v_1
v_2
Questions & Answers
QUESTION:
Suppose that is a standard normal random variable and that \(Y_{1}\) and \(Y_{2}\) are \(\chi^{2}\)-distributed random variables with \(v_{1}\) and \(v_{2}\) degrees of freedom, respectively. Further, assume that Z, \(Y_{1}\) and \(Y_{1}\) are independent.
a Define \(W=Z / \sqrt{Y_{1}}\). Find \(E(W)\) and \(V(W)\). What assumptions do you need about the value of \(v_{1}\)? [Hint: \(W=Z\left(1 / \sqrt{Y_{1}}\right)=g(Z) h\left(Y_{1}\right)\). Use Theorem 5.9. The results of Exercise 4.112(d) will also be useful.]
b Define \(U=Y_{1} / Y_{2}\). Find \(E(U)\) and \(V(U)\). What assumptions about \(v_{1}\) and \(v_{2}\) do you need? Use the hint from part (a).
Equation Transcription:
Text Transcription:
Y_1
Y_2
chi_2
v_1
v_2
Y_1
Y_2
W=Z/ sqrt Y_1
E(W)
V(W)
v_1
W=Z(1/ sqrt Y_1)=g(Z)h(Y_1)
U=Y_1/Y_2
E(U)
V(U)
v_1
v_2
ANSWER:
Solution :
Step 1 of 2:
Let Z is a standard normal random variable and and
are
distributed random variables and
and
are degrees of freedom.
Our goal is:
a). We need to define and we need to find E(W) and V(W).
b). We need to define U = and we need to find E(W) and V(W).
a).
Now we need to define and we need to find E(W) and V(W).
We assume that Z, , and
are independent.
Property standard normally distributed variable is
f(z)=
E(z) = = 0
The expected value is the sum of the product of each probability with its probability is
Using result from exercise 112d is
We use the theorem from 5.9 and that and
are independent is
E(W)=
Therefore .
And variance of V(W)=
V(W) =
V(W) =
Therefore, V(W) is