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Let $x_a$, $x_b$, and $x_c$ be three random samples from a PDF $f(x)$. The samples are then sorted into $x_1$, $x_2$, and $x_3$ in ascending order. How do I get the distribution of $x_1$, $x_2$, and $x_3$? Is there a generalization for this for more samples?

I have solved the case for $x_1$ and $x_3$ (or $x_n$ in the general case) with min and max convolutions, but I can't make it work with the intermediate cases. My attempt was $P(x_2)\ \alpha \ P(x_b|x_a<x_b)P(x_b|x_b<x_c)$ then normalizing it, but it doesn't seem to hold up when I manually solved it on the case of a fair dice.

zvxayr
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1 Answers1

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Community wiki answer so the question can be marked as answered:

As noted in the comments, these are order statistics. If the cumulative distribution function corresponding to $f$ is $F$, the cumulative distribution functions for the $x_i$ are

$$ F_i(x)=\sum_{j=i}^3\binom3jF(x)^j(1-F(x))^{3-j}\;, $$

so

\begin{eqnarray*} F_1(x) &=& 1-(1-F(x))^3\;, \\ F_2(x) &=& (3-2F(x))F(x)^2\;, \\ F_3(x) &=& F(x)^3\;. \end{eqnarray*}

joriki
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