Please help with e,f,g and h :

Assume that individual lifetime 7 follows a Geometric distribution: P(T = x) = (1 - 1)34, for x = 1, 2,.... with 0 " = 1- 11=0 (b) Show that the mortality rate (x) = P(7 = x|7 2 x) does not depend on the age, i.c., A(x) = 1 for all x. (c) Write down the likelihood and log-likelihood function given the observation (7;). (d) Show that the maximum likelihood estimation A, is given by n I'M a Ti (e) Show that 1, is a (weak) consistent estimator of 1. (f) Show that the observed Fisher Information /(1) is given by = () n + (1 -1)2 S(. -1). (g) Find an estimate of the variance Var(1,). (h) Use the following sample dataset of 7 to answer the questions below. 13, 15, 8, 6, 5, 25, 7, 8, 35, 41 i. Calculate A,, and give a 95% confidence interval of 1. ii. Give an estimate' S (x) of the survival probability of individual at the age of 45