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( yk-? The Erlang distribution is a probability distribution for non-negative real numbers. It models the time taken for k events to occur, under the condition that the events occur at the same rate over time. It finds extensive use in telecommunications and medical research. The version that we will look at has a probability density function of the form p(ylv, k) = exp(-e-"y- kv) (1) (k-1)! where e* = exp(x) and Y e R7, i.e., y can take on the values of non-negative real numbers. In this form it has two parameters: the “shape" k > 0, which is the number of events we are interested in waiting for, and v ER, which is the log-inverse-rate (also called a log-scale) of the distribution, which controls the rate at which the events we are modelling occur. Often k is not treated as a learnable parameter, but is rather set by the user depending on the context. If a random variable follows an Erlang distribution with log-inverse-rate v and shape k we say that Y ~ Er(v,k). If Y ~ Er(v,k), then E [Y] = ke" and V [Y] = ke2u 1. Produce a plot of the Erlang probability density function (1) for the values y E (0,15), for (v = 0, k = 1), (v = 1, k = 1) and (v = -1/2, k = 2). Ensure that the graph is readable, the axis are labelled appropriately and a legend is included. [2 marks] 2. Imagine we are given a sample of n observations y =(41, ..., yn). Write down the joint probabil- ity of this sample of data, under the assumption that it came from an Erlang distribution with log-inverse-rate v and shape k (i.e., write down the likelihood of this data). Make sure to simplify your expression, and provide working. (hint: remember that these samples are independent and identically distributed.) [2 marks] 3. Take the negative logarithm of your likelihood expression and write down the negative log- likelihood of the data y under the Erlang model with log-inverse-rate v and shape k. Simplify this expression. [1 mark] 4. Derive the maximum likelihood estimator û for v, under the assumption that k is fixed; that is, find the value of v that minimises the negative log-likelihood, treating k as a fixed quantity. You must provide working. [2 marks] 5. Determine expressions for the approximate bias and variance of the maximum likelihood esti- mator û of v for the Erlang distribution, under the assumption that k is fixed. (hints: utilise techniques from Lecture 2, Slide 22 and the mean/variance of the sample mean) [3 marks]