Solution Found!
Suppose that Y1, Y2, and Y3 are independent ?
Chapter 5, Problem 155SE(choose chapter or problem)
Problem 155SE
Suppose that Y1, Y2, and Y3 are independent χ 2-distributed random variables with ν1, ν2, and ν3 degrees of freedom, respectively, and that W1 = Y1 + Y2 and W2 = Y1 + Y3.
a In Exercise 5.87, you derived the mean and variance of W1. Find Cov(W1, W2).
b Explain why you expected the answer to part (a) to be positive.
Reference
Suppose that Y1 and Y2 are independent χ 2 random variables with ν1 and ν2 degrees of freedom, respectively. Find
a E ( Y 1 + Y2).
b V ( Y 1 + Y2). [Hint: Use Theorem 5.9 and the result of Exercise 4.112(a).]
Questions & Answers
QUESTION:
Problem 155SE
Suppose that Y1, Y2, and Y3 are independent χ 2-distributed random variables with ν1, ν2, and ν3 degrees of freedom, respectively, and that W1 = Y1 + Y2 and W2 = Y1 + Y3.
a In Exercise 5.87, you derived the mean and variance of W1. Find Cov(W1, W2).
b Explain why you expected the answer to part (a) to be positive.
Reference
Suppose that Y1 and Y2 are independent χ 2 random variables with ν1 and ν2 degrees of freedom, respectively. Find
a E ( Y 1 + Y2).
b V ( Y 1 + Y2). [Hint: Use Theorem 5.9 and the result of Exercise 4.112(a).]
ANSWER:
Solution:
Step 1 of 2:
Let ,
, and
are independent
- distributed random variable with
,
, and
degrees of freedom, respectively and that
=
+
and
=
+
-
The claim is to find the Cov(
,
)
We know that V() = V(
+
)
= 2+ 2
V() = V(
+
)
= 2+ 2
Similarly, V(+
) = V(2
+
+
)
= 4V() + V(
) + V(
)
= 8 + 2
+ 2
Then, Cov(,
) =
=
= 6 + 2
+ 2
Hence, Cov(,
) = 6
+ 2
+ 2