Random Variable

Study guides: Exam P-1, Exam FAM, Exam ALTAM, Exam MAS-I

A Random Variable $X$ is a function that assigns a real number to each outcome in a Sample Space $S$ , enabling numerical analysis of random experiments.

$X : S \to \mathbb{R}$

$X$ is discrete if its possible values form a finite or countably infinite set, and continuous if its probabilities come from a density, $P(a \le X \le b) = \int_a^b f(x)\,dx$ — so its CDF has no jumps and every single value has probability 0. See Discrete Random Variable and Continuous Random Variable.
The two classes do not cover every case. A payment under a Deductible, $Y = (X - d)_+$ with $X$ continuous, takes a continuum of values yet puts probability $P(X \le d) > 0$ on the single value 0: it is a mixed random variable, neither discrete nor continuous (see Payment Random Variable).
They are fully characterized by their probability distribution, which describes how probability is spread across their possible values

Defining a Random Variable for Coin Flips Two fair coins are flipped. Define a random variable $X$ as the number of heads. List the values $X$ can take and their probabilities.

Answer The sample space is $S = \{HH, HT, TH, TT\}$ , each with probability $1/4$ . The random variable $X$ maps: $X(TT) = 0,\quad X(HT) = X(TH) = 1,\quad X(HH) = 2$ So the distribution is $P(X=0)=\tfrac{1}{4}$ , $P(X=1)=\tfrac{1}{2}$ , $P(X=2)=\tfrac{1}{4}$ .