**Example to a Maximum Likelihood estimation**

## Example to a Maximum Likelihood estimation

Let us take *N* number of y_{i} measured datapoint, that are indepedent but were produced by the same process which can be described by a normal distribution with expected value, and variance.

The joint probability density function of the independent y_{i} events is the product of the individual pdf's:

The log-likelihood function reads:

We now look for the maximum by differentiating and setting the result equal to zero:

from that our estimation for is:

the average of the measured data. The partial derivatives according to :

from that:

For completeness we mention that the formula for the variance we still have an unknown , and we may get tempted to estimate it by . As it is well known this would result in a biased estimate, that can be corrected by a factor of *N/(N-1)*.

Now we proceed with estimation theory and the Bayesianestimators.