## Question

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.

## Solution

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Since this is a normal distribution, we know that,

(1)

(1) is the probability density function (PDF) of a Gaussian distribution.

We have data points . The joint probability density of observing n data points:

(2)

Taking logs of (2),

(3)

Simplifying (3),

(4)

To find MLE of a certain parameter, we have to differentiate w.r.t the desired parameter. Here, the parameter is .

(5)

* Thus, (5) is the desired MLE. *