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Consider a random variable X with associated probability density function given by: f(x) = θ^-2xe^-x/θ for x > 0 and f(x) = 0 otherwise

a) Assume you collected a dataset {X1,…,Xn} what is the associated likelihood function (explain all your intermediate steps and assumptions to derive the likelihood function)

b) Find the maximum likelihood estimator, θ^ for θ

c) calculate the value for θ^ if the dataset is given by { 0.25, 0.75, 1.50, 2.50,2}

d) what is the interpretation of θ^ in terms of the distribution for X (consider the maximum likelihood estimator for θ^)

a) To derive the likelihood function, we start by writi...