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.