In Mathematica, how can I define an arbitrary probability distribution? - wolfram-mathematica

Question

I want an arbitrary function p[x] that integrates to 1 and for all x, 0 <= p[x] <= 1. Some kind of transformation rule?

Answer 1

You could use ProbabilityDistribution for this together with an undefined function of x:
dist = ProbabilityDistribution[p[x], {x, -Infinity, Infinity}];
It now knows a few rules to apply:
continuous probability density: probability of a single value is zero
In[26]:= Probability[x == 0, x \[Distributed] dist]
Out[26]= 0
the probability of having a value at all
In[28]:= Probability[x > 0 || x <= 0, x \[Distributed] dist]
Out[28]= 1
The CDF at - infinity
In[29]:= CDF[dist][-\[Infinity]]
Out[29]= 0
The CDF at + infinity
In[30]:= CDF[dist][\[Infinity]]
Out[30]= 1
The PDF
In[32]:= PDF[dist][x]
Out[32]= p[x]
However, it doesn't assume the PDF of the distribution is normalized:
In[33]:= Integrate[PDF[dist][x], {x, -Infinity, Infinity}]
Out[33]= Integrate[p[x], {x, -Infinity, Infinity}]
The latter can be taught, defining an UpValue for p:
p /: Integrate[p[x], {x, -Infinity, Infinity}] = 1;
Now it can integrate the PDF:
In[4]:= Integrate[PDF[dist][x], {x, -Infinity, Infinity}]
Out[4]= 1
You know that your second requirement, i.e. 0 <= p[x] <= 1, is not generally true for probability density functions, do you?

Answer 2

In case you're just asking for examples of density functions (PDFs) that match your criteria, here are two (out of uncountably many):
p(x) = 1 if 0 < x < 1
0 otherwise
p(x) = x/2 if 0 < x < 2
0 otherwise
We could even generalize those slightly:
p(x) = 1/k if 0 < x < k
0 otherwise
p(x) = 2x/k^2 if 0 < x < k
0 otherwise
The latter works for k >= 2.
We can even generalize that with another parameter to get a class of such functions with arbitrary exponent
p(x) = (a+1)/k^(a+1)*x^a if 0 < x < k
0 otherwise
which works for all a > 1 and k > a+1.
For more interesting examples I think you'll need to give more criteria.
You mention a transformation rule so perhaps you'd like to take an arbitrary bounded function on R1 and translate/scale it so that it's always between 0 and 1 and integrates to 1.
That will have a straightforward answer as long as you can get the min, max, and integral of the given function.
Go ahead and edit the question to ask that if that's indeed what you're looking for.