1). On a busy Sydney y crossroad , traffic accidents happen almost every week . The distribution of the number of accidents in a week is believed to be Poisson. Then the numbers of accidents observed over a period of a weeks would be a sample of size n of X = (X,, X2. .... X,,) where the X; are i.id. random variables, cach with distribution f (r, A) = 272-45 7 , I = 0, 1, 2 , ... a) Construct a simple unbiased estimator of the probability r(A) = bed 2 of exactly two ac- cidents in a week. (You may utilise the indicator (2 (* ) in your construction). b) Use any argument to show that T= X, is complete and minimal sufficient for A. c) Derive the UMVUE of r(A) = Ale- d ) Calculate the Cramer -Rao bound for the minimal variance of an unbiased estimator of T (A). Does the variance of the UMVUE estimator of T(A) attain this bound ? Give reasons for your answer. 2). Let X1, X2,.... X, be i.i.d. random variables with f(r; 0) = e -I, if x 2 0 0 being 0 else unknown parameter. a) Find the cumulative distribution function Fx (y;0) and the density fx (y; 0) of the smallest of the observations, *(1). Hint : you could use P(X() y, X2 > y, . .., X, 2 9) =1 - [P(X , 2 y)]". b) Find EX() c) Show that X() is complete . Find the UMVUE of 0. 3). The discrete random variable X takes values according to one of the following distribution types (0 c 9 = (0, 0.1)): P(X = 1) P(X = 2) P(X = 3) P(X = 4) Distribution 20 20 205- 04 1+ 0 - 40 - 203 Distribution II 0 - -03 202 1 - 20 - 202 + 03 In each of the two cases determine whether the family of distributions {Po(X)], 0 c e is complete. Give reasons. 4) High cholesterol levels can cause hearth disease. Let X;, i = 1, 2, ...,n be a random indicator variable with X; = 1 if the ith patient has a hearth discase and X, = 0 otherwise . If z; is the