1. [11 marks} State whether each of the following statements is TRUE of FALSE, and give a reason for your answer. PrlEA} {a} If two events A and B are independent, then PrIIAIB} = Frtei' (b) An unbiased estimator is always better titan a biased estimator. (e) A Bayesian 95% credible interval for a parameter species an interval within which the parameter lies with probability {1.95. (d) lfthe expected value of X is l and the standard deviation of' X is 2, then the expected value and standard deviation of 5):.\" + T are 12 and m respectively. {e} The maximum likelihood estimator is always a mction of a sufcient statistic. (f) If two random variables X and Fare independent then the correlation between X and 1' is G. (g) A ll-of'xa condence interval for a parameter :9 species that in repeated sampling the percentage of such intervals that cover the true value ofti will converge to 95%. (h) The maximum likelihood estimator is always unbiased and consistent. [i] The power of a statistical test is inversely related to the sample size. [j] A type 1 error corresponds to reaching a decision to reject the null hypothesis when it is in fact true. {it} The pvaiue of' a hypothesis test is the probability of obtaining a test statistic value at least as large as the one observed in the sample. [3 marks] A random variable X has probability density functionx) = its: for l] s: x E 3?. and some constant k. and ffx) = ii otherwise. (a) Show that k = {15. (b) What is the means: and variance er2 of X? Now let XL X1 ... X\" be independently and identically distributed random variables with the density anction dened above. The sample mean is J? =ii1=1 XI . (c) Find E [If] and ForUT]. ._ .F" {a} Show thatX - 1;. (e) Use the Central Limit Theorem to nd an approximation for ice m2 r1 :2 14a) (=1