The idea

Continuous random variables

Probability density functions — the two validity conditions, probabilities as areas, E(X) and Var(X) by integration, the median and percentiles from cumulative area, the mode as the maximum of f, and the continuous uniform distribution as the simplest special case.

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Statistics · Statistical distributions

Continuous random variables

Why it works

Probability as area

A continuous quantity — a waiting time, a mass, a length — can take any value in an interval, so no single value can carry a lump of probability:

\mathrm{P}(X = a) = 0

for every exact

a

. Instead, probability is spread across the interval with a probability density function

\mathrm{f}(x)

, and probability lives in areas:

\mathrm{P}(a < X < b) = \int_a^b \mathrm{f}(x)\,\mathrm{d}x.

Because single points carry no probability,

\mathrm{P}(X \leqslant a)

and

\mathrm{P}(X < a)

are the same — strict versus non-strict makes no difference for a continuous variable.

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