Maximum of:       Triangular, Uniform and half-Halo           New in mathStatica 2.0

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

Let  MaxMin_2.gif   with pdf   f(x),   
    let   MaxMin_3.gif  with pdf g(y), 
    and let Z half-Halo with pdf h(z):

In[1]:= MaxMin_4.gif

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:

In[2]:= MaxMin_6.gif

Out[2]:= MaxMin_7.gif


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


Maximum of:       Six different Uniform random variables           New in mathStatica 2.0

Let MaxMin_11.gif with pdf f(x):

In[1]:= MaxMin_12.gif

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

In[2]:= MaxMin_13.gif

Out[2]= MaxMin_14.gif

Problem: Find the pdf of MaxMin_15.gif.  

Solution: The solution pdf is simply:

In[3]:= MaxMin_16.gif

Out[3]= MaxMin_17.gif

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

In[4]:= MaxMin_18.gif


Here is a plot of the solution pdf:

In[5]:= MaxMin_19.gif

Out[5]= MaxMin_20.gif

A quick Monte Carlo 'check’ of the exact solution we have just plotted:
We first generate 6 pseudo-random data sets corresponding to MaxMin_21.gif, and each containing 250000 pseudo-random drawings:

In[6]:= MaxMin_22.gif  

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 MaxMin_23.gif. Then, we map the Max function across each sample of 6, generating our 250000 empirical drawings of the sample maximum:

In[7]:= MaxMin_24.gif


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

In[8]:= MaxMin_26.gif

Out[8]= MaxMin_27.gif

For the win!


Minimum of:       Pareto and Exponential           New in mathStatica 2.0

Let X ∼ Pareto(a, b)  with pdf f(x):

In[1]:= MaxMin_30.gif

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

In[2]:= MaxMin_31.gif

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

Solution: The solution is simply:

In[3]:= MaxMin_32.gif

Out[3]= MaxMin_33.gif