Distribution of a sum of normal random variables

Question

Let $X_1, X_2, ..., X_n$ be independently distributed variables where $X_k \sim N(k\mu, 1)$ for $k = 1, 2, ..., n$ and $\mu\in\mathbb{R}$ unknown.

I calculated that the ML estimator for $\mu$ is equal to $$\hat{\mu}_{ML}=\frac{\sum_{k=1}^n k x_k}{\sum_{k=1}^n k^2}.$$ My question is, what is the distribution of $\hat{\mu}_{ML}$.

Any help is greatly appreciated.

Answer 1

The numerator (being a sum of Gaussian) is distributed $$ \mathcal N(\sum k^2\mu_k,\sum k^2) $$while the denominator is simplified as $$ \sum^n_{k=1}k^2=\frac{n(n+1)(2n+1)}{6}=c $$So the distribution you need is$$ \mathcal N(\frac{1}{c}\sum k^2\mu_k,\frac{1}{c^2}\sum k^2) $$

Answer 2

The answer above is only partially correct. Your estimator is indeed normally distributed, with the mean exactly $\mu$ (so your estimator is unbiased) and the variance $$ \left(\frac1{\sum_{k=1}^n k^2}\right)^2\times\sum_{k=1}^n k^2\cdot 1=\frac1{\sum_{k=1}^n k^2}. $$