Continuous Random Variables

• A probability density function satisfies f(x) ≥ 0 and ∫f(x) dx = 1 over its domain.
• P(a < X < b) = ∫_a^b f(x) dx; P(X = a) = 0.
• E(X) = ∫x f(x) dx; Var(X) = ∫x² f(x) dx − [E(X)]².
• Median m: ∫ from the lower limit to m of f(x) dx = 0.5. Other percentiles similarly.
• Domains may be infinite, e.g. f(x) = 3/x⁴ for x ≥ 1.

Area is probability

A pdf f(x) is defined over a single interval and is zero elsewhere. It is never negative, and the total area under it is 1 — use this to find an unknown constant. Probabilities are areas, so it makes no difference whether inequalities are strict. For an infinite domain, use the limit: ∫₁^∞ x⁻⁴ dx = [−x⁻³/3]₁^∞ = 1/3.

The sketch helps: the mode is where f(x) is greatest.

Integrals for everything

The mean is ∫x f(x) dx and the variance is ∫x² f(x) dx minus the square of the mean. The median m splits the area in half: solve ∫ f(x) dx from the lower limit to m = 0.5. The upper quartile q satisfies an area of 0.75 to its left (or 0.25 to its right — often easier for an infinite domain). The cumulative distribution function itself is not required; just integrate the area directly.