Continuous variables and probability density

When the possible values form a continuous interval, it no longer makes sense to assign a probability to each single value: probability is described by a density function, and it becomes an area.

Definition — Probability density function

A continuous r.v. $X$ is described by a probability density function $f(x)\ge 0$ with $\int_{-\infty}^{+\infty}f(x)\,dx = 1$. The probability that $X$ falls in an interval is $P(a\le X\le b) = \int_a^b f(x)\,dx.$

Remark

For a continuous variable, $P(X=x_0) = 0$ for every single value $x_0$ (the integral over a point is zero). It only makes sense to ask for the probability of an interval.

The total area under the curve is always $1$: it is the continuous analogue of the condition $\sum p_i = 1$ of discrete distributions.

Links

Topics: Distribuzioni probabilita
Concepts: Densita di probabilita · Variabile aleatoria
Skills: Interpretare grafico