Final answer:
The MLE of μ for a normal distribution with known variance σ2 is the sample mean, found by summing all sample observations and dividing by the number of observations.
Step-by-step explanation:
Finding the MLE of μ Given σ2 is Known
When you have a random sample Y1, ..., Yn from a normal distribution with mean μ and variance σ2, the Maximum Likelihood Estimator (MLE) of μ is the sample mean, provided that σ2 is known. This is because the likelihood function for μ, given the observed data, is maximized when μ is equal to the average of the sample means.
To find the MLE of μ, you sum up all the observed values Yi and divide by the number of observations, n. The formula for the MLE of μ is:
μMLE = (∑ Yi) / n
This is under the Central Limit Theorem which dictates that the distribution of sample means will tend to be normally distributed. Thus, the sample mean is the best estimate for μ when the population variance is known.