Example sheet 4. Statistical Modelling: Mathematical Tripos, Part IIC Summary: Example sheet 4. Statistical Modelling: Mathematical Tripos, Part IIC P.M.E.Altham, Statistical Laboratory, University of Cambridge. January 18, 2005 These questions are all adapted from recent Part IIA Tripos questions. There may be some overlap between questions, eg on basic bookwork matters. You are not intended to do all these questions for 1 supervision. Be selective. Keep some questions for later revision purposes. 1998/4/13M (long question) Suppose that Y 1 ; :::; Yn are independent random variables, and that Y i has probability density function f(y i j i ; ) = exp[(y i  i b( i ))= + c(y i ; )]: Assume that E(Y i ) =  i , and that there is a known link function g such that g( i ) = T x i ; where x i is known and is unknown: Show that (a) E(Y i ) = b 0 ( i ), (b) var(Y i ) = b 00 ( i ) = V i say, and hence (c) if `( ; ) is the log-likelihood function from the observations (y 1 ; :::; yn ) then @`( ; ) @ = Collections: Mathematics