please answer question 1 part (d) and question 2 part (c)

1. Let X, be independent Poisson random variables with parameter A for i = 1, 2, . . ., n. (a) Write out the joint distribution for X = (X1, ..., Xn) as an exponential family and specify its components. (b) Write out the likelihood and log-likelihood functions for A. (c) Show that the maximum likelihood estimate A is the sample mean X = LieIX, n (d) It is often desirable for the mean and variance of a random variable to be unrelated (e.g., normal distribution). This is clearly not true for the Poisson, but a suitable transformation could ameliorate this problem. Suggest and justify an estimator for the transformation h(A) = vx, and determine its asymptotic distribution including the variance. 2. In addition to ordered 2 x k tables, the y statistic and test can be used for a 2 x 2 table. (a) Using the notation in the notes, simplify y and its variance for the 2 x 2 table. (b) Yule (1912) proposed a correlation measure for the 2 x 2 table as a function of the odds ratio . Q = - 1+ 1 Show that y = Q for the 2 x 2 table. (c) Use the delta method for a function of y and the var(y) from the notes to derive a variance formula for the odds ratio v. (d) The following data was collected for a study on the distances motor vehicles pass a cyclist. Parked Cars Distance No Yes > 1m 10135 7344