Question

15 Random vanables ( 21, 2,... xs} are taken from a normally distributed Population with mean U and Varrance 6² - Assuming that they are all indopondant. Wnte down an expression of thes is rev which has the following pdfs. (0 N(0, 1) (ii) X266) (1) F (6,7) (1) T (4)

Answer 1

1.

N(0,1), i = 1, 2, ... 15 o

2.

$U=\sum_{i=1}^{6}Y_i^2=\sum_{i=1}^{6}\frac{(x_i-\mu)^2}{\sigma^2}\sim \chi^2(6)$

3.

let

$V=\sum_{i=7}^{13}Y_i^2=\sum_{i=7}^{13}\frac{(x_i-\mu)^2}{\sigma^2}\sim \chi^2(7)$

w U/6 7 ΣΕ(– μ)? V/7 6 Σ2, (α; – μ)2 F(6, 7)

4.

let

$S=\sum_{i=2}^{5}Y_i^2=\sum_{i=2}^{5}\frac{(x_i-\mu)^2}{\sigma^2}\sim \chi^2(4)$

$T=\frac{Y_1}{\sqrt{S/4}}=2*\frac{(x_1-\mu)}{\sqrt{\sum_{i=2}^{5}(x_i-\mu)^2}}\sim T(4)$