3 questions from my statistics assignment, answers for any of them would be helpful

[1 5 marks] 5. The probability function of a binomial(n, p) random variable is f(x) = (")p(-p)" for= =0,1,....n and the moment generating function is M() = [(1 - p) + pel]". (a) Show that the exponentially tilted probability function (with parameter () is binomial(n, p) and give the expression for p. (b) Let S = El, X, where X, are lid binomial(20, 0.6) random variables. Use the exponential tilt for each X, to construct the importance sampling estimator of P(S > 140). Give the optimal value of t to use in the tilt. (c) Use Matlab's built-in binomial random variable generator to simulate the X's which are then used to construct simulated values of S. i. Use crude Monte Carlo to estimate P(S > 140) by simulating n = 100,000 observations of S. Provide a point estimate and a 95% confidence interval. ii. Use the importance sampling estimator to estimate P(S > 140) by simulating n = 100,000 observations of S. Provide a point estimate and a 95% confidence interval. ifi. Compute the efficiency of importance sampling over crude Monte Carlo. [10marks] 6. State whether each of the following statements is True or False. Briefly discuss. (a) Consider estimating the derivative of a function f at the point a using simulation estimates of f in a finite-difference-type estimate of the derivative, Ma) = [ath) - fa) h where h > 0 is a small fixed real number and /(x) is a simulation estimate of /(2)). i. The use of common random numbers is an effective way to reduce the variance of f(x) compared with crude Monte Carlo. ii. Using common random numbers in f(x) gives an unbiased estimator of f(x). (b) It is not possible to use the method of inversion to simulate Normal(a, o?) random variables. [10 marks] 7. Suppose that (X, Y ) are uniformly distributed in a circle of radius a having joint probability density function fxy(zy) = c forosity' sa o otherwise where c is some constant that makes fxy a valid joint density function. (a) Find c. (b) Let R be the distance from the centre of the circle to (X, Y). Find the cumulative distribution function of R. (c) Show that R is a uniform(0, a?) random variable