Categorical Distribution#
The Categorical distribution is the most general discrete distribution and is parameterized by a vector \(p\) where each element \(p_i\) specifies the probabilities of each possible outcome.
Key properties and parameters#
Support |
\(x \in \{0, 1, \ldots, |p|-1\}\) |
Parameters:
\(p\) : (array) Probabilities of each category, \(p_i \geq 0\) and \(\sum_i p_i = 1\).
Probability Density Function (PDF)#
\[
f(x) = p_x
\]
Cumulative Distribution Function (CDF)#
\[\begin{split}
F(x \mid p) = \begin{cases}
0 & \text{if } x < 0 \\
\sum_{i=0}^{x} p_i & \text{if } 0 \leq x < |p| \\
1 & \text{if } x \geq |p|
\end{cases}
\end{split}\]
where \(p\) is the array of probabilities for each category.
See also
Related Distributions:
Bernoulli - The Categorical distribution is a generalization of the Bernoulli distribution to more than two outcomes.
Discrete Uniform - A special case of the Categorical distribution where all outcomes have equal probability.