Discrete vs Continuous — Section 4: Random Variables & Distributions

Random variables come in two main flavors based on the kind of values they can take.

Discrete random variables take values in a countable set — the integers, a finite list, etc. Their distribution is captured by a probability mass function (PMF):

p(x) = P(X = x), \qquad \sum_x p(x) = 1

Examples: number of heads in 10 flips (Binomial), number of arrivals in an hour (Poisson), score on a multiple-choice quiz.

Continuous random variables take values in an interval or in all of $\mathbb{R}$ . For a continuous variable, $P(X = x) = 0$ for every single point — there are uncountably many of them. Instead, probability is described by a probability density function (PDF) $f(x)$ , and probabilities are areas under that curve:

P(a \le X \le b) = \int_a^b f(x)\, dx

Examples: a stock's daily log return, the time between trade arrivals, the height of a randomly chosen person.

Some real-world quantities are mixed — an option payoff is exactly $0$ most of the time (a discrete atom) and continuous when in-the-money. The full theory handles both with measure theory, but in practice you can usually treat them piecewise.

Random variables come in two main flavors based on the kind of values they can take.

Discrete random variables take values in a countable set — the integers, a finite list, etc. Their distribution is captured by a probability mass function (PMF):

p(x) = P(X = x), \qquad \sum_x p(x) = 1

Examples: number of heads in 10 flips (Binomial), number of arrivals in an hour (Poisson), score on a multiple-choice quiz.

P(a \le X \le b) = \int_a^b f(x)\, dx

Examples: a stock's daily log return, the time between trade arrivals, the height of a randomly chosen person.