(Solved):
Consider a random variable X with associated probability density function given by: f(x)=2xe ...
Consider a random variable X with associated probability density function given by: f(x)=θ−2xe−θf, for x>0 and f(x)=0 otherwise. a) Assume you collected a dataset {X1inis ,Xn}, 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