Continuous Probability Distributions
A probability distribution is a function that gives the probability of occurrence of a random event. It can be either discrete or continuous, depending on the domain. We can formally define it as:
Here can determine the distribution as either discrete or continuous as being a discrete or continuous set, respectively.
There are a couple of useful ways to describe a probability distribution:
- Cumulative Distribution Function (CDF)
- Probability Density Function (PDF)
Cumulative Distribution Function (CDF)
The CDF of a distribution is the value it will take for less than (or equal to) the given value (generalized as ). Formally,
Note: Since probabilities are always non-negative, CDF is a non-decreasing function.
Probability Density Function (PDF)
More often, we are interested in calculating the value for a particular rather than the whole limit. Here we can use the Probability Density Function (PDF). This can be calculated by taking the derivative of the CDF:
The expected value of a random variable is an intuitive measure of its mean, which is formally defined as:
As a simple example, suppose the probability of a dice roll is as follows:
The expected value will be:
This notion of expected value can be extended to variance, etc., which are collectively known as moments.
The moment can be defined as:
The first four moments are:
Usually, we consider only mean and variance.
Now we’ll revise some commonly used probability distributions along with their respective JAX functions.