Definition

Because there is an infinite continuum of possible values $x$ for a continuous random variable $X$, the probability of $X$ being exactly equal to a particular value is zero $\Pr(X=x)=0$ ! Therefore, the approach used to define the probability distribution of discrete random variables can not be used to describe the distribution of continuous random variables. Instead, the probability distribution of a continuous variable is defined by the probability of a random variable being less than or equal to a particular value

$$\Pr(X\leq x) = F(x)$$ (4.4)

The probability distribution function, $F(x)$, is close to zero for large negative values of $x$ and increases towards one for large positive values of $x$.

The probability distribution function is sometimes referred to more specifically as the cumulative distribution function (c.d.f). The probability of a continuous random variable $X$ being in a small interval $(a,a+\delta x]$ is given by

$$\Pr(a < X \leq a+\delta x) = F(a+\delta x)-F(a)\approx \left[\frac{dF}{dx} \right]_{x=a}\delta x$$ (4.5)

The derivative of the probability distribution, $f(x)=\frac{dF}{dx}$, is known as the probability density function (p.d.f.) and can be integrated with respect to $x$ to find the probability of $X$ being in any interval

$$Pr(a < X\leq b) = \int_a^b f(x)dx=F(b)-F(a)$$ (4.6)

In other words, the probability of $X$ being in a certain interval is simply given by the integrated area under the probability density function curve.