#squared-loss

Question 1: Single choice

Samples $x_1, \dots, x_N$ come from $\mathcal{N}(\mu, 1)$ with unknown mean $\mu$ . Which statement about the maximum likelihood estimate of $\mu$ is correct?

It is the sample mean, as the negative log-likelihood is $\frac{1}{2}\sum_i (x_i - \mu)^2$ plus a constant1
It is the sample median, because the Gaussian density is symmetric about $\mu$ 2
It is the sample mean only for large $N$ ; for small samples the estimate is the most frequent value3
It cannot be found without first estimating the variance from the same data4

Timed test