I have posted essentially the same question on reddit, but it has been some time now, and I have not yet received an answers, so I am also posting it here.
I have five measurements of speed $v_i\pm\sigma_i$, where i ranges from 1 to 5, and all the values of $v_i$ and $\sigma_i$ are different. The measurements are assumed to be independent of each other and normally distributed around a mean speed $v$. We can write a likelihood function for these measurements that has the following form:
$$L = \prod_1^5 \frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{1}{2}(v_i-v')^2/(\sigma_i)^2}$$
There are two things that I am trying to figure out how to extract from this expression: The maximum likelihood estimator $v'$ for $v$, and the $\sigma(v')$ uncertainty associated with it. Now, it is fairly easy to extract the estimator $v$ from this: simply take the $\log(L)$, differentiate with respect to $v'$, and set the derivative to zero. Then solve for $v'$. You get something like $v' = \frac{\sum_1^5 \frac{v_i}{σ_i^2}}{\sum_1^5 \frac{1}{σ_i^2}}$.
But I am stuck with the standard deviation. The answers to this question say that the correct expression for $\sigma(v')$ is $\sigma(v')^2=\left(\sum_1^5 \frac{1}{\sigma_i^2}\right)^{-1}$ but for the love of me I could not figure out how this expression was obtained.
