Derive Maximum Likelihood Estimate (MLE) for the mean of a univariate normal distribution. Assume N samples independently drawn from a normal distribution with known variance and unknown mean . Show all intermediate steps and assumptions.
Since this is a normal distribution, we know that,
(1) is the probability density function (PDF) of a Gaussian distribution.
We have data points . The joint probability density of observing n data points:
Taking logs of (2),
To find MLE of a certain parameter, we have to differentiate w.r.t the desired parameter. Here, the parameter is .
Thus, (5) is the desired MLE.