Answer to Question #271410 in Statistics and Probability for hope

Question

b) Let 𝑌1,𝑌2,...,𝑌𝑛 be a random sample from a population with probability density

function in part a). Show that the best test for the hypothesis in part a) rejects 𝐻0

if

∏𝑦𝑖 ≤𝑐

𝑛

𝑖=1

where 𝑐 solves the probability equation

𝛼 =𝑃(∏ 𝑦𝑖 ≤𝑐𝑛𝑖=1 |𝜃 =2). (8)

c) Let 𝑋1,𝑋2,...,𝑋𝑛 be a random sample from 𝐺𝐴𝑀𝑀𝐴(2,𝛽) distribution, and

consider 𝑌=∑ 𝑋𝑖𝑛𝑖=1 .

Show whether or not 𝑌 is a pivotal quantity and give its distribution.