## Example: Product of Pareto and Uniform New in **mathStatica 2.0**

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

In[1]:=

and let random variable *Y* ∼ Uniform(α, β) with pdf *g*(*y*):

In[2]:=

Problem: Find the pdf of *V* = *X Y*, denoted *h*(*v*).

Solution: The solution is simply:

In3]:=

Out[3]=

A quick Monte Carlo ‘check’ of the exact solution we have just derived:

It is always a good idea to check that a theoretical solution that has been derived is consistent with pseudo-random data. To illustrate, let us suppose that (*a*=2, *b*=3, α=1, β=4). To perform a quick Monte Carlo check, we first generate 100000 pseudo-random drawings of X, and 100000 pseudo-random drawings of Y:

In[4]:=

We now make a frequency plot of the product of the X and Y pseudo-random data, and then compare the pseudo-random Monte Carlo solution (—) with the theoretical symbolic solution *h*(*v*) (—) derived above:

In[5]:=

Out5=

Looks good!

## Example: Product of two Standardised Normals New in **mathStatica 2.0**

Let random variable *X* ∼ *N*(0,1) with pdf *f*(*x*):

In[1]:=

The pdf of the product of two standardised Normals can then be elegantly derived via:

In[2]:=

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## Example: Product of Two Triangulars New in **mathStatica 2.0**

Let random variable *X* ∼ Triangular(-1/2, 1, 2) with pdf *f*(*x*):

In[1]=

and let random variable *Y* ∼ Triangular(-1, 2, 3) with pdf *g*(*y*):

In[2]=

The following diagram plots the pdf of both *f*(*x*) (—) and *g*(*y*) (—):

Problem: Find the pdf of *V* = *X***Y* (i.e. the pdf of the product of the two random variables).

Solution: Here is the solution pdf, say *h*(*v*):

In[3]:=

Out[3]=

In[4]:=

The solution has a piecewise form. Here is a plot of the solution pdf:

In[5]:=

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

It is always a good idea to check that a theoretical solution that has been derived is consistent with pseudo-random data. Here, we generate 100000 pseudo-random drawings of X and 100000 pseudo-random drawings of Y:

In[7]:=

We can now make a frequency plot of the product of the X and Y pseudo-random data, and then compare the pseudo-random Monte Carlo solution (—) with the theoretical symbolic solution *h*(*v*) (—) derived above:

In[8]:=

Out[8]=

For the win!