Question 5 - Maximum Likelihood Estimation (15 Marks) The exponential distribution Is a probability distribution for non-negative real numbers. It is often used to model walting or survival times. The version that we will look at has a probability density function of the form p(ylo) = exp( 2 'p- () where y Ky le., y can take on the values of non-negative real numbers. In this form It has one parameter. a log-scale parameter . If a random variable follows a gamma distribution with log-scale u we say that Y ~ Exp(u). If Y ~ Exp(o). then A[Y'] = e" and v[Y] = c.". Question 5.a (4 Marks) Imagine we are given a sample of n observations y = ('i, .... ";). Write down the joint probability of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter 17 (Le, write down the likelihood of this data). Make sure to simplify your expression, and provide working. YOUR ANSWER HERE Question 5.b (2 Marks) Take the negative logarithm of your likelihood expression and write down the negative loglikelihood of the data y under the exponential model with log-scale v. Simplify this expression YOUR ANSWER HERE Question 5.c (4 Marks) Derive the maximum likelihood estimator o for v. That Is. find the value of w that minimises the negative log-likelihood. You must provide working. YOUR ANSWER HERE Question 5.d (5 Marks) Determine the approximate blas and varlance of the maximum likelihood estimator p of , for the exponential distribution. Note that this le a challenge question, only attempt if you are comfortable with your progress. You should not use consultations to ask about this question as the tutors will proritize answering queries about other questions. YOUR ANSWER HERE