Random variables: pmf, pdf, cdf, sf and quantiles

A random variable assigns a number to each outcome; its distribution is described by a pmf or pdf, a cdf, a survival function and a quantile function.

F(x) = P(X \le x), \qquad S(x) = P(X > x) = 1 - F(x)

Cumulative distribution function (cdf) and survival function (sf)

$X$: the random variable
$F(x)$: probability of a value at most $x$ (lower tail)
$S(x)$: probability of a value strictly greater than $x$ (upper tail). For discrete variables $S(k) = P(X > k)$ , so $P(X \ge k) = S(k-1)$ .

\E(X) = \sum_x x\,p(x) \;\text{ or }\; \int x\,f(x)\,dx, \qquad \Var(X) = \E\big[(X-\mu)^2\big] = \E(X^2) - \mu^2

Mean and variance of a distribution

$p(x)$: probability mass function (discrete): $p(x) = P(X = x)$
$f(x)$: probability density function (continuous): areas under it are probabilities
$\mu$: the mean $\E(X)$

Q(p) = \min\{x : F(x) \ge p\}

Quantile (inverse cdf)

$p$: a probability between 0 and 1
$Q(p)$: the smallest value whose cdf reaches $p$ (for continuous distributions, the value with exactly $p$ below it)

When to use

Whenever a quantity is uncertain before it is observed: a count of defects, a waiting time, a test statistic.
Discrete variables (counts) have a pmf: probabilities sit on separate values. Continuous variables (measurements) have a pdf: probabilities are areas, and $P(X = x) = 0$ for any single value.

How to interpret

The cdf answers "at most" questions; the sf answers "more than" questions; the quantile function answers "which value cuts off this probability?" (critical values, percentiles).
A pdf value is a density, not a probability; it can exceed 1.
Engine convention: for discrete distributions sf(k) = P(X > k) (strictly greater, as in R and SciPy) and quantile(p) is the smallest k with P(X ≤ k) ≥ p. Tails are computed directly, never as 1 − cdf, so tiny p-values keep their precision.

Common mistakes

For a count, $P(X \ge 3) = 1 - P(X \le 2) = S(2)$ , not $S(3)$ . "At least" and "more than" differ by one value.
For a continuous variable, $\le$ and $<$ give the same probability; for a discrete one they do not.
Reading a density height as a probability.

Example

A fair die: $p(x) = 1/6$ for $x = 1, \dots, 6$ . $F(2) = 2/6 \approx 0.333$ , $S(4) = P(X > 4) = 2/6$ , mean $= 3.5$ , variance $= 91/6 - 3.5^2 \approx 2.917$ .