Discrete Random Variables 1

Jan 17, 2024

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Terminology

Informally, a \(\textbf{random variable}\) is a quantity \(X\) whose value depends on some random event. The \(\textbf{space (or range)}\) of \(X\) is the set \(S\) of possible values of \(X\). If this set \(S\) is finite or countable (i.e., can be listed as a sequence \((x_1. x_2, \cdots\)), the random variable is called \(\textbf{discrete}\).

General formulas

• \(\textbf{Probability mass function (p.m.f):}\)
  • \(f(x) = \mathbb{P}(X = x)\) for all \(x \in S\)
  • \(f(x) \geq 0\) and \(\underset{x \in S}{\sum} f(x) = 1\)
  • \(\textbf{Uniform distribution}\) on a set \(S\): Each of the values \(x \in S\) has the same probability, i.e., \(f(x) = 1/n\) for each value \(x\), where \(n\) is the number of values.
• \(\textbf{Expectation (mean):}\)
  • \(\mu = \mathbb{E}(X) = \underset{x \in S}{\sum} x \cdot f(x)\)
  • \(\mathbb{E}(c) = c, \:\: \mathbb{E}(cX) = c\mathbb{E}(X), \:\: \mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)\)
  • \(\textbf{Expectation of a function of } X: \mathbb{E}(u(X)) = \underset{x \in S}{\sum} u(x)f(x)\)
• \(\textbf{Variance:}\)
  • \(\sigma^2 = \text{Var}(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2\)
  • \(\text{Var}(X) = \mathbb{E}((X - \mu)^2)\)
  • \(\text{Var}(c) = 0, \:\: \text{Var}(cX) = c^2\:\text{Var}(X), \:\: \text{Var}(X + c) = \text{Var}(X)\)
  • \(\textbf{Standard deviation}: \sigma = \sqrt{\text{Var}(X)}\)
• \(\textbf{Moment-generating function:}\)
  • \(M(t) = \mathbb{E}(\mathcal{e}^{tX}) = \underset{x \in S}{\sum} \mathcal{e}^{tx} f(x)\)
  • The derivatives of \(M(t)\) at 0 generate the moments of \(X: M^{\prime} (0) = \mathbb{E}(X), M^{\prime\prime}(0) = \mathbb{E}(X^2), \:\: M^{\prime\prime\prime} (0) = \mathbb{E}(X^3),\) etc.