1. Question 1. Consider a sample of n :5 2 independent and identically distributed random variables Y1, . . . ,1; from an expl} distribution with probability density function fly: Al = lexel-ly} y :r G A :- i}. 1 We will use the notation H = J.' a Let Y1 = min 1'31...1Y be the smallest observation in the sample, and this { 3 sample is collected from a distribution where fy[y) and F32 {y} are the probability density function and the distribution function respectively. Show that Fully} = 1 - [1 - Fvlivll\" Hint: Evaluate Pfl} :3- y}. {b} Show that Fill has probability density function {pdf}: fruity} = n-[1 - Fviy)]\"'lfv{yJ- {c} Hence show that Fill follows an exp[n}L) distribution if the sample is from an exp} distribution. [d] Show that the following two estimators are both unbiased estimators of 3: $1 = Eff-El]. Ell-Cl d2 = ? where ? is the sample mean. Hint: You may need to use common probability distributions PDF on iLearn. [e] Derive the relative efciency of 31 compared to d2. 1'3'"hich estimator do you prefer? [f] Determine whether ll and [3'3 are consistent estimators of H. {g} Write down the likelihood function and the loglikelihood function, as functions of A. {h} Find a suicient statistic for A. [i] Derive the maximum likelihood estimator {MLE} of A. Comment on your result in relation to the previous part.