Consider three different distributions defined over three different domains of support …

Let with pdf *f*(*x*),
let with pdf *g*(*y*),
and let *Z* ∼ half-Halo with pdf *h*(*z*):

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Here are the three pdf’s illustrated:

**Suprematism No. 1:** *f*(·) Triangular *g*(·) Uniform *h*(·) half-Halo

We seek the pdf of *W* = max(*X*, *Y*, *Z*) . The solution pdf is simply:

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Here is a plot of the pdf of the maximum, together with the underlying pdf’s:

**Suprematism No. 2:** The Malevich Maximum — pdf of the maximum, together with the 3 underlying pdf’s

Let with pdf *f*(*x*):

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Here are the six pdf’s corresponding to *b* = 1, 2, 3, 4, 5 and 6:

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Problem: Find the pdf of .

Solution: The solution pdf is simply:

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with domain of support: (we define piecewise functions over the real line)

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Here is a plot of the solution pdf:

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A quick Monte Carlo 'check’ of the exact solution we have just plotted:

We first generate 6 pseudo-random data sets corresponding to , and each containing 250000 pseudo-random drawings:

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Next, we transpose from 6 data sets each of size 250000 … to … 250000 samples each of size 6. Each sample of 6 represents a single pseudo-random drawing from . Then, we map the Max function across each sample of 6, generating our 250000 empirical drawings of the sample maximum:

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We can now make a frequency plot to compare the pseudo-random Monte Carlo solution (—) with the theoretical symbolic solution φ(*x*) (—) derived above:

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For the win!

Let X ∼ Pareto(a, b) with pdf *f*(*x*):

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and let Y ∼ Exponential(λ) with pdf *g*(*y*):

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Problem: Find the pdf of *W* = min(*X*, *Y*).

Solution: The solution is simply:

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