DISCRETE PROBABILITY DISTRIBUTION

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Discrete Probability Distribution

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◇Overview

A Discrete Probability Distribution specifies all possible values in the countable support $R_X = \{x_1, x_2, \dots\}$ of a discrete random variable $X$ together with their probabilities $p(x_i) = P(X = x_i)$. Common parametric families include the Bernoulli, Binomial, Geometric, Hypergeometric, and Poisson distributions.

◇Key Definitions

• Validity Conditions: (1) $p(x_i) \ge 0$ for all $x_i \in R_X$, and (2) $\sum_{x_i \in R_X} p(x_i) = 1$.
• Cumulative Distribution Function (CDF): $F(x) = P(X \le x) = \sum_{t \le x} p(t)$, a right-continuous step function with jumps of height $p(x_i)$ at each support point $x_i$.
• Expected Value (Mean) & Variance:
  $\mu = E[X] = \sum_{x} x \, p(x), \qquad \sigma^2 = \text{Var}(X) = \sum_{x} (x - \mu)^2 p(x) = E[X^2] - (E[X])^2$

◇Worked Example

Problem: Let $X$ have distribution $P(X = 0) = 0.2$, $P(X = 1) = 0.5$, and $P(X = 2) = 0.3$. Compute $E[X]$ and $\text{Var}(X)$.

Solution:

• $E[X] = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1$.
• $E[X^2] = 0^2(0.2) + 1^2(0.5) + 2^2(0.3) = 0 + 0.5 + 1.2 = 1.7$.
• $\text{Var}(X) = E[X^2] - (E[X])^2 = 1.7 - (1.1)^2 = 1.7 - 1.21 = 0.49$.

◇References & Navigation

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• Related Notes: ◇Random Variable, ◇Hyper-geometric Distribution