Exercise 3. Consider an exponentially distributed random variable f (X; ) = A exp(-x). (a) Show that the expected value of X is 1-3. (b) Verify that the estimator = 1/7 is biased, with bias equal to fact that E = Xi n-1 What happens as n o? i=1 [5] n-1 Hint: you may use the [5] (c) Simulate n values of x and calculate the mean of x = 1x at values of n = {1,..., 15}. i=1 n Repeat 100 times for each value of n and hence use these samples to estimate the bias of the estimator = 1/X. You can generate such samples using the code below: lambda.hat-matrix(NaN, 15, 100) 1 2 for (n in 1:15) { 3 for (j in 1:100) { 1 1 1 1 1 1The reason for this is to consider the marginals of a Multinomial({1, 2, 3, 4, 5, 6}, { , , , , , }) or to use the property of conditional Binomials: if (X + Y) Binomial (n, p) for some p then we can clearly say that (Y|X + Y) ~ ~ Binomial(X + Y, ) since the 6s are as probable as the 5s. Then conditional probability rules tell us that Y ~ Binomial(n, p) and we already know Y ~ Binomial (n, 1/2) so p must be 13. Easiest of all is to simply consider that rolling a 5 or a 6 is with probability - + = and assume this is Binomial. 4 } } x=rexp(n,rate=5) lambda.hat [n, j]=1/mean(x) Here we have used the sample mean of as an estimator of its expected value and we've used = 5 (but the result will hold for any ). Now plot the bias against n and confirm that it follows the equation A. Comment on your result. n-1 [5] 56 N 7