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Consider a random variable $X$ with associated probability density function given by: $f(x)=θ_{−2}xe_{−θf},forx>0$ and $f(x)=0$ otherwise. a) Assume you collected a dataset ${X_{1_{inis}},X_{n}}$, what is the associated likelihood function (explain all your intermediate steps and assumptions to derive the likelihood function)? [3 marks] b) Find the maximum likelihood estimator, $θ^$ for $θ$. [10 marks] c) Calculate the value for $θ$ if the dataset is given by ${0.25,0.75,1.50,2.50,2}$. [2 marks] d.) What is the interpretation of $α$ hat in terms of the distribution for $X$ ? Hint: consider the maximum likelihood estimator of $α_$hat $=(∑(i=1)_{∧}n×i)/(2n)$ [10 marks]

a) To derive the likelihood function, we need to find the joint probability density function (pdf) for the dataset {X?, X?, ..., X?}. Since the random variable X has a probability density function given by:

f(x) = 0, otherwise The joint probability density function (pdf) can be expressed as the product of individual densities for each observation: Substituting the given probability density function: Taking the product and rearranging, we get:

Refer above solution