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.)
Step 1 of 1:
Let be a random sample from a normal distribution with mean and .
The density for X is
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
We are applying the logarithm likelihood function is
We compute the first partial derivatives with respect to mean and variance and we set equal to 0.
When and Then the second partial derivatives is
Here Xand also given random sample with size n.
Then the the maximum likelihood estimator of mean and variance are and respectively.