Problem 10E

Let X1,..., Xn be a random sample from a N(µ, σ2) population. Find the MLEs of µ and of σ. (Hint: The likelihood function is a function of two parameters, µ and σ. Compute partial derivatives with respect to µ and σ and set them equal to 0 to find the values and that maximize the likelihood function.)

Solution :

Step 1 of 1:

Let be a random sample from a normal distribution with mean and .

The density for X is

f(x)=

Our goal is:

We need to find the maximum likelihood estimator of mean and variance .

Now we have to find the maximum likelihood estimator of mean and variance .

Here the likelihood function for the sample is

L(,)=

=

=

We are applying the logarithm likelihood function is

Ln L(,)=-nln.

We compute the first partial derivatives with respect to mean and variance and we set equal to 0.

Then,

=

Here

=

= 0

Then,

=

=

Here =0

= 0

When and Then the second partial derivatives is

=and

=

=

=

Here Xand also given random sample with size n.

Then the the maximum likelihood estimator of mean and variance are and respectively.