Suppose X1,...,xn iid Exponential() is a set of n observations drawn independently from an Exponential distribution with rate parameter 1. (a) Write out the likelihood function. (b) Write out the log-likelihood function. (c) Find the score function by taking the partial derivative of the log-likelihood function. (d) Set the score function equal to zero and solve for the parameter 1. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down. (g) You want to estimate how long it takes you to do each of the lecture assignments for STATS67. You find that the first 7 assigments took you (33, 76, 23, 56, 71, 88, 12) minutes. Assuming that the time it takes you to complete the assignments follows an Exponential Distribuiton with the same rate parameter 1, what is the Maximum Likelihood Estimator for the mean time that it takes you to complete a lecture assignment?
