STATS 100B Homework 4 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
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.
2022-02-15