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STATS 100B Homework 4 2022
发布时间:2022-02-15
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STATS 100B Homework 4
2022
In all the problems, we define
n
n = n !Xi ,
n
S = n − 1 !(Xi −
n)2 .
Problem 1 (20 pts)
Let X1 , X2 , . . . , X16 be i.i.d. from N(50, 102). Find
a. Find the distribution of "
1 (Xi − 50)2 . (10 pts)
b. Find the distribution of "
1 (Xi −
16)2 . (10 pts)
Problem 2 (10 pts)
Let n and S
denote the sample mean and sample variance of an independent random sample of size n from N(0, σ2). Find the distribution of
.
Problem 3 (10 pts)
Let X1 , . . . , Xn be i.i.d. from N(µ1 , 1) and Y1 , . . . , Ym be i.i.d. from N(µ2 , 1). Xi’s and Yi’s are independent. Find the distribution of W = "1 (Xi −
n)2 + "
1 (Yi −
m)2 .
Problem 4 (20 pts)
Let X1 , . . . , X18 be i.i.d. from N(µ1 , 7σ2), and Y1 , . . . , Y23 be i.i.d. from N(µ2 , 3σ2). Xi’s and Yi’s are independent. S =
"
1 (Xi −
18)2 and S
=
"
1 (Yi −
23)2 . For what value of c does the expression c
have the F17,22 distribution?
Problem 5 (20 pts)
Let X1 , . . . , Xn be i.i.d. from N(0, σ2). n =
"
1 Xi , and S
=
"
1 (Xi −
n)2 .
a. Find c such that c
S has the F distribution. (10 pts)
b. What are the numerator and denominator degrees of freedom of the F distribution? (10 pts)
Problem 6 (10 pts)
Please prove the following statement.
If X ∼ Fm,n , then ∼ Fn,m . (10 pts)
Problem 7 (10 pts)
If T follows a t7 distribution, find t0 such that (a) P(|T| < t0) = 0.9 and (b) P(T > t0) = 0.05.