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(Solved): Consider a random variable X with associated probability density function given by: f(x)=2xe ...

Consider a random variable with associated probability density function given by: and otherwise. a) Assume you collected a dataset , 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, for . [10 marks] c) Calculate the value for if the dataset is given by . [2 marks] d.) What is the interpretation of hat in terms of the distribution for ? Hint: consider the maximum likelihood estimator of hat [10 marks]

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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:

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