Notes:Distribution of the sample median

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Problem overview

Let $X_1,\ldots,X_{2m+1}$ be a sample from a population $X$, meaning that the $X_i$ are i.i.d random variables, for some $m\in\mathbb{N}_{0}$. We wish to find:

• $$\P{\text{Median}(X_1,\ldots,X_{2m+1})\le r}$$ - the Template:Cdf of the median.

Initial work

Since the variables are independent then any ordering is as likely as any other (which I proved the long way, rather than just jumping to $$\frac{1}{(2m+1)!}$$ - silly me) however the result, found in Probability of i.i.d random variables being in an order and not greater than something will be useful.