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## Distinguishing two normal distributions given 1 vs. n samples

Let $S$ be a vector of length $n$ where each entry follows a Normal distribution. We know that one entry (say, at some index $s$ which we don't know) follows $\mathcal{N}(E_c,V_c/m)$ while every other entries follows $\mathcal{N}(E_w,V_w/m)$. Assume $E_c > E_w$. We want to know what the probability is that an entry with index $\neq s$ is bigger than $S_s$. That is, the probability that $S$ tells us what $s$ is.

As far as I can tell we can give a lower bound on this probability by looking at the difference distribution of $\mathcal{N}(E_c,V_c/m)$ and $\mathcal{N}(E_w,V_w/m)$, i.e., ask when samples from $\mathcal{N}(E_c-E_w,V_c/m + V_w/m)$ are $> 0$. Then, we can ask what the probability is that this happens $n-1$ times, i.e., take this probability $p$ to the power of $n-1$.

However, this seems to be too pessimistic. We are only sampling $\mathcal{N}(E_c,V_c/m)$ once and not $n-1$ times, how do I account for this?

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