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+ 100% 1. (a) Let X = (X1, Am.... Xa) be a random sample from a normal distribution with mean fi and variance of 1. If pa is known and o' is unknown, show that E(X, - p) is a sufficient statistic for a, Show all steps of your working. it If fu and a are both unknown, show that X (Le. the sample mean) and 5' (Le. the sample variance) are jointly sufficient for p and of (b) A family of discrete random variables has support r = (1,2,3,4) and a parameter space of # = (2,4.6,8), with the following probability mas functions: Px (5;0 = 2) 0.25 1.25 0.25 Px (50 =4) 0.00 0.35 0.35 0.30 Px (5; ( = 6) 0.30 0.20 0.20 0.30 Px (5;0 - 8) 0.30 0.35 0.05 0.30 For each possible observed sample x of size n = 1, provide the maximum likelihood estimates of 0. Justify your