Suppose that a random variable $X \in [0,1]$ is drawn according to the density $f(x|\theta)$ conditional on the realization of $\theta \in [0,1]$. From $x$, we can generate $c$ according to some rule $c=g(x)$. We have know that $g$ is one-to-one. This procedure will give a distribution of $c$ given a realization of $\theta$. Now consider a median of $c$ and how the median of $c$ changes when $\theta$ changes. Under what condition on $f$ can it be that the median of $c$ is distinct for each $\theta$ (or at least there is only a measure zero set of $\theta$ that will give the same median of $c$.
How does the median change when the underlying variable changes
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probability
statistics