- A Question Let X Be A Random Variable With A Normal Distribution N M1 02 And Y A Random Variable With A Normal Distr 1 (145.77 KiB) Viewed 21 times
)a Question : Let X be a random variable with a normal distribution N(M1,02), and Y a random variable with a normal distribution N(H2, oż). Consider a random sample of size nį from X and a random sample of size n2 from Y. Assume that X and Y are independent. = (a) Assuming that oỉ = oź = oʻ, show that the maximum likelihood estimator for o? is (nı – 1) sî + (12 – 1)s ni + n2 ô2 where 2-(X; - X) ni - 1 Σ2, (Υ; - Y)2 si and s 1 n2 - 1 (b) Assuming that oỉ + ož, show that the maximum likelihood estimators for oị and oź are: ô = (n1 – 1)sí ni (n2 – 1)s ôz = n2 (c) Let Lo be the likelihood function evaluated at the maximum likelihood estimators of o?, M1 and uz in part (a) and Ly be the likelihood function evaluated at the maximum likelihood estimators of o, o,Mi and uz in part (b). To test the hypothesis H. : 01 = 02 against H1 :01 + 02, we use the likelihood ratio Lo/Lj. Show that (@?)1/2(62)^2/2 A= (@2)(ni+n2)/2 A=