1. ## Maximum likelihood estimates

Find the maximum likelihood estimates for $\displaystyle \theta_{1}=\mu$ and $\displaystyle \theta_{2}=\sigma^{2}$ if a random sample of size 15 from $\displaystyle N(\mu, \sigma^{2})$ yielded the following values:

31.5
35.2
31.6
36.9
29.6
36.7
33.8
34.4
35.8
30.1
30.5
34.5
33.9
34.2
32.7

Could anyone show me the steps in doing this problem? It would really help me understand this chapter and hopefully tackle other similar problems. Thanks

2. The MLEs for a normal distribution when you don't know both the mean or the variance is...

$\displaystyle \hat\theta_1=\bar X={\sum_{i=1}^{15}X_i\over 15}$

and

$\displaystyle \hat\theta_2=\hat\sigma^2={\sum_{i=1}^{15}(X_i-\bar X)^2\over 15}$

(that's n, not the unbiased estimator, with n-1).

To prove that, these are the MLEs, you obtain the likelihood function which is just the product of the 15 densities.
You should take the log and then differentiate wrt each of these two parameters to obtian this.