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

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?