Is there a general formula for calculating distribution of the maximum of the minimum of random variables?

For example: say I have independent random variables $X_1,X_2,X_3$ distributed with CDFs $F_1,F_2,F_3$ respectively. I'm interested in the distribution of $Z = max(min(X_1,X_2),min(X_1,X_3),min(X_2,X_3))$. My current solution is to calculate $P(Zleq t) = P({min(X_1,X_2)leq t} land {min(X_1,X_3) leq t} land {min(X_2,X_3)leq t })$.

I can evaluate this by using a modified version of the inclusion-exclusion principle in terms of simpler expressions. This creates an expression with 55 terms that can be simplified to 5. Is there a theorem to go straight to the simplified 5 term expression? I'd like to generalize this to the case of the maximum of an arbitrary number of minimums of an arbitrary subset of independent variables. Using my current algorithm with the inclusion-exclusion principle doesn't scale as well as I would like. Is there a better solution to this problem?

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#### Best Answer

Let $X_1, X_2, X_3$ be continuous independent *non-identical* random variables. We have a sample of 3 values, namely $(X_1, X_2, X_3)$, where:

- 1 value is drawn from $X_1$,
- 1 value is drawn from $X_2$ and
- 1 value is drawn from $X_3$.

We seek the pdf of:

$$Z = max(min(X_1,X_2),min(X_1,X_3),min(X_2,X_3))$$

Without loss of generality, imagine that the sample is such that $X_1 < X_2 < X_3$. Then, $Z = max(X_1,X_2) = X_2$ (*i.e.* we seek the pdf of the second largest order statistic, from the sample of 3 values).

In summary: given sample $(X_1, X_2, X_3)$ of non-identical random variables, we seek the pdf of the $2^{nd}$ order statistic.

This problem is solvable exactly, but, for any typical example (with overlapping domains of support) the computation can be difficult to do by hand, and it is easiest to solve with the help of a computer algebra system. See, for instance:

- Rose, C. and Smith, M.D. (2005), Computational order statistics,
*The Mathematica Journal*, 9(4), 790-802.

**Example**

Let:

- $X_1 sim text{Exponential}(1)$ with pdf $f(x_1)$
- $X_2 sim N(0,1) quad quad quad$ with pdf $g(x_2)$
- $X_3 sim text{Uniform}(-1,1) $ with pdf $h(x_3)$

That is:

Here is a plot of the 3 parent pdf's:

We seek the pdf of $Z$, namely the pdf of the $2^{nd}$ order statistic in sample of size 3, where 1 value is taken from each of ${f,g,h}$. This can be calculated with the `OrderStatNonIdentical`

function from the *mathStatica* package for *Mathematica*:

Here is a plot of the pdf of $Z$:

**Monte Carlo check**

Here is quick check of the empirical pdf of $Z = max(min(X_1,X_2),min(X_1,X_3),min(X_2,X_3))$ using Monte Carlo methods:

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