Question

Multi-part question:

Let X1, ..... , Xn be random variables that describe the height of students from a class, in the logarithmic
scale.

A) Write the statistical model (there might be more than one suitable distribution).

B) Assume that X1, ... ,Xn form a random sample from the normal distribution with known mean
θ and unknown variance σ^2 . Find the maximum likelihood estimator of the variability of the
height (in log scale) of the students, this is, find the maximum likelihood estimator of σ^2.

C) Under the assumptions in (B), find the maximum likelihood estimator of the second population
moment, this is, find the maximum likelihood estimator of E(X^2).

D) Under the assumptions in (B), show that the maximum likelihood estimator of σ^2 is a sequence
of consistent estimators for the variance of the height of students, σ^2.
Hint: Use the law of large number for the random variables (X1 - θ)^2, ... ,(Xn - θ)^2 and
compute their expectation.

Answer 1