Homework 2
B07703076 財金五 王鈺翔
1. If Z is a random variable such that Z follows a standard normal distribution, then
𝑍 2 follows 𝑥 2 (1).
2. If for 1 ≤ 𝑖 ≤ 𝑘, 𝑌𝑖 are random variables following 𝑥 2 (1), then ∑𝑘𝑖=1 𝑌𝑖 follows
𝑥 2 (𝑘).
3. If Z is a random variable such that Z follows a standard normal distribution and Y
is a random variable such that Y follows 𝑥 2 (𝑘), then
𝑍
√𝑌
𝑘
follows t(k).
4. If 𝑌1 is a random variable such that 𝑌1 follows 𝑥 2 (𝑘1 ), 𝑌2 is a random variable
such that 𝑌2 follows 𝑥 2 (𝑘2 ), then
𝑌
( 1)
𝑘1
𝑌
( 2)
𝑘2
follows F(𝑘1 , 𝑘2 )
1
5. Let 𝑆𝑛2 = ∑𝑘𝑖=1(𝑋𝑖 − 𝑋̅𝑛 )2 , Z = 𝑛 2 (𝑋̅𝑛 − 𝜇)/𝜎,
Y = 𝑆𝑛2 /𝜎 2
We know that Z and Y are independent, Y has the 𝑥 2 distribution with n-1
degrees of freedom and Z has the standard normal distribution.
By definition, 𝑈 =
𝑍
𝑌
(𝑛−1)1/2
has the t distribution with n-1 degrees of freedom.
We can thus rewrite U as
1
𝑛 2 (𝑋̅𝑛 − 𝜇)
𝑈=
𝑆2
(𝑛 −𝑛 1)1/2