A continuous random variable is modeled over a continuous range of values. Instead of assigning probability to individual points, its distribution is described by a probability density function (PDF).

A density $f_X(x)$ satisfies

$$f_X(x)\ge0$$

and

$$\int_{-\infty}^{\infty}f_X(x),dx=1.$$

Probability is area under the density

For an interval $[a,b]$,

$$P(a\le X\le b)=\int_a^b f_X(x),dx.$$

Thus probability is represented by area under the density curve.

Density is not probability at a point

For a continuous model,

$$P(X=x)=0$$

for every individual value $x$. This does not mean the value is impossible; it means a single point has zero width and therefore zero area.

For the same reason,

$$P(a<X<b)=P(a\le X\le b).$$

Including or excluding finitely many endpoints does not change the probability.

Example: constant density

If

$$f_X(x)=\frac12\qquad 0\le x\le2$$

and zero elsewhere, then

$$P(0.5\le X\le1.5) =\int_{0.5}^{1.5}\frac12,dx =\frac12.$$

The height of a density describes probability per unit of the variable; only an integral over a range gives probability.