Discrete vs Continuous

Countable outcomes versus a continuum of possible values.

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.