Question

10.95 Suppose that we have a random sample of four observations from the density function e-y/0, y > 0, f(y(0) = { elsewhere. Find the rejection region for the most powerful test of Ho : 0 = 0o against H:0 = 0a, assuming that a > 00. [Hint: Make use of the x2 distribution.] Is the test given in part (a) uniformly most powerful for the alternative 0 > 0.? a b

Can you explain why we multiply this by 4 times? You dont have the answer the questions in the picture. I just need to know why it's 4.

Answer 1

Note that: We have a random sample of 4 observations.

Since they are random samples , they are identical and independent.

Let, be the four random samples.

Y1, 42, 43, 44

Note that,

Y1, 42, 43, YA = 41, 42, 43, 44

By definition:

likelihood of 0 : L(0) = product of pdf of 4 random samples

So,

L(0) = f(y)f(420) f(430) f(440)
( since,
Y1, 42, 43, 44
are independent)

(133-) * (*11),() - [(4-) dro (30) (45) 11 - (914) S [I = (0)7 T=2