Discrete Probability Space

A discrete probability space is a probability_space with a countable sample space.

Definition

Let $\Omega = \{\omega_1, \omega_2, \ldots\}$ be a countable set and $(p_n)_{n \in \mathbb{N}}$ be a sequence of real numbers $p_n \in [0,1]$ such that $\sum_{n \in \mathbb{N}} p_n = 1$.

The triplet $(\Omega, \mathcal{P}(\Omega), \mathbb{P})$ is a discrete probability space, where:

• $\Omega$ is the sample space
• $\mathcal{P}(\Omega)$ is the power set (all subsets of $\Omega$)
• $\mathbb{P}$ is the discrete_probability_measure defined by $\mathbb{P}(A) = \sum_{n: \omega_n \in A} p_n$

Examples

• Fair coin flip: $\Omega = \{H, T\}$, $p_H = p_T = \frac{1}{2}$
• Fair die roll: $\Omega = \{1, 2, 3, 4, 5, 6\}$, $p_i = \frac{1}{6}$
• Geometric distribution: $\Omega = \mathbb{N}$, $p_n = (1-p)^{n-1}p$

Related

• probability_space
• discrete_probability_measure
• countable