Exercise 3. Consider an exponentially distributed random variable f(X; ) = exp(x). (a) Show that the...

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Exercise 3. Consider an exponentially distributed random variable f(X; ) = exp(x). (a) Show that the expected value of X is 1/1. [5] (b) Verify that the estimator = 1/ is biased, with bias equal to ^. Hint: you may use the fact that E = 1. What happens as n o? 1 i=1' xi n-1 (c) Simulate n values of x and calculate the mean of x = 11; at values of n = Zi=1 n [5] {1,..., 15}. 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: 1 lambda. hat-matrix (NaN, 15, 100) 2 for (n in 1:15) { 3 for (j in 1:100) { 4 5 6 7 } 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 2-1. Comment on your result. n-1 [5] Exercise 3. Consider an exponentially distributed random variable f(X; ) = exp(x). (a) Show that the expected value of X is 1/1. [5] (b) Verify that the estimator = 1/ is biased, with bias equal to ^. Hint: you may use the fact that E = 1. What happens as n o? 1 i=1' xi n-1 (c) Simulate n values of x and calculate the mean of x = 11; at values of n = Zi=1 n [5] {1,..., 15}. 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: 1 lambda. hat-matrix (NaN, 15, 100) 2 for (n in 1:15) { 3 for (j in 1:100) { 4 5 6 7 } 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 2-1. Comment on your result. n-1 [5]