Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y,...,Y, for a random sample that is, n independent observation, of the exponential distribution Exp (H) . The parameter > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi a.) Derive the joint density function f(y; ) specifying the model. b.) Derive the likelihood function L () and the log likelihood function 1 () . c.) Define the maximum likelihood estimate (y) in the model for the parameter and form the maximum likelihood estimator ^ (Y) in the model for the parameter . d.) Assume that n = 6 and the observed data (i.e., six dog encounter intervals (min)) is y = (Y1, Y2, ..., 95, 96) = (2.25, 8.40, 5.10, 11.85, 3.30, 8.10) .Use the data to calculate the ML estimate for the expected value of the intervals. Also consider what can be deduced from this: On the basis on this analysis of problem, do you think it is possible, i.e. likely, that expected value would actually be 3 (minutes). e.) Calculate the expected value E (Y) of the ML estimator ^ (Y) Tell using this (with justification) is the ML estimator unbiased? (Use the definition of unbiased estimator) f.) Calculate the variance var^ (Y) of the ML estimator ^ (Y)