Create a distribution from a set of points on a curve

Situation

We have a set of co-ordinates that we wish to use to construct a distribution:

1. {x, f(x)} for a continuous distribution where f(x) is (or is proportional to) the probability density at value x;

2. {x, F(x)} a continuous distribution where F(x) is the cumulative probability (P(X<=x)) at value x; or

3. {x, p(x)} for a discrete distribution where p(x) is (or is proportional to) the probability of value x.

Uses

There are many uses of this technique. For example:

• Converting the results of a constructed Bayesian inference calculation into a distribution; or
• Constructing a spliced or mixed distribution by averaging, or manipulating probability density functions for the component distributions; or
• knowing the pdf or pmf of a specific distribution not available in your software

Application

We can use the same techniques as explained in Method 3 to create distributions from a set of points:

If the data set is of the form of {x, f(x)}, we can use a General distribution

If the data set is of the form {x, F(x)}, we can use the Cumulative distribution 

If the data set is of the form {x, p(x)}, we can use the Discrete distribution

The {x} values must be in ascending order for the General and Cumulative distribution because they construct a distribution shape. For the Discrete distribution this is unnecessary because it is simply a list of values.