Question

Suppose {X 1​ ,⋯,X n​ } is a random sample chosen from the distribution with pdf f(x)=(θ−1)

ˣ⁻ θ for x>1 where θ>1. (a). Find the M.L.E.
θ^ of θ. (b). Let τ= 1- for x>1 where θ>1.

Answer 1

The question requires calculating the maximum likelihood estimator for a given pdf and references the Central Limit Theorem in relation to random sampling and the distribution of sample means.

Step-by-step explanation:

The question pertains to finding the maximum likelihood estimator (MLE) for the parameter θ of a probability density function (pdf), and also touches on the Central Limit Theorem as it relates to random sampling and the distribution of sample means. The random variable X follows a specific distribution with a given pdf, f(x)=(θ-1) ˣ⁻ θ for x>1 where θ>1. Part (a) asks for the calculation of the MLE, θ^, and part (b) mentions a transformation τ but seems incomplete.

Proceeding to identify the solutions to the implied tasks: for part (a), calculating the MLE involves setting up the likelihood function based on the pdf, taking the logarithm to find the log-likelihood, then finding the derivative with respect to θ, and solving for θ by setting the derivative equal to zero. The Central Limit Theorem for Sample Means suggests that if X is a random variable with a known or unknown distribution, the distribution of the sample means (ΣX/n) will approximate a normal distribution as the sample size n increases.