# Maximum Likelihood Estimation

Let the random samples x1, x2, x3,... xn drawn from normal distribution with known mean zero

a. Construct the likelihood function

b. Find the Nature Log of the likelihood function

c. Find the best estimate of the variance by MLE

d. What is the relationship between this estimate and the second moment <x²>

e. If we use second moment <x²> to estimate the variance, is the estimation biased? Why?

f. Under conditional (e) can this estimate be used to estimate population variance without bias? Why?

https://brainmass.com/statistics/normal-distribution/maximum-likelihood-estimation-279716

#### Solution Preview

Let the random samples x1, x2, x3,... xn drawn from normal distribution with known mean μ which is equal to 0 and unknown variance φ.

F(x,φ)=[1√2П φ] exp [-(x- μ)²/(2 φ)]

a. Construct the likelihood function L(x, φ)

b. Find the Nature Log of the likelihood function ln L(x, φ)

c. Find the best ...

#### Solution Summary

In this solution the maximum likelihood estimate of the variance of a normal distribution with mean zero is derived. It has been shown that this estimate is an unbiased estimate