Solved – Maximum of minimum of random variables

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: