Homework 2

Problem 1

Proof the theorem from the class: Let \(X \sim \chi^2(n), Y \sim \chi^2(m)\) and independent. Let \(Z=(X/n)/(Y/m)\). Then \(Z \sim F(n,m)\)

Problem 2

Say \(X_1,...,X_n\) have a geometric rate p distribution and are independent. Let \(M=\max\{X_1,...,X_n\}\). Find the density of M.

Problem 3

Show that if \(X\sim t(n)\), then \(X^2\sim F(1, n)\).

Problem 4

Below is the data from a Binomial distribution Bin(10, p), already in the form of a table:

x	Freq
0	11
1	38
2	60
3	47
4	26
5	10
6	6
7	2

Draw the log-likelihood function as a function of p.