2.4 Continuous random variables

To describe random variables that can take any value on the real line (e.g. the discrepancy between two length measurements, or the logarithm of the concentration of a chemical), or on an interval of the real line (e.g. the time until a traffic light changes from red) we need slightly different mathematical tools than we used for discrete random variables. The starting point, however, is the same as for discrete random variables; the CDF is

Figure 2.2 (Link) shows the cdf for a continuous random variable.

The key attribute is that this is a continuous function; there are no jumps. Since $F_X(x)$ is continuous everywhere, $𝖯\left(X=x\right)=F_X(x)-{lim}_{i\to \mathrm{\infty }}{F}_X(x-1/i)=0$ for all $x$. Hence, also,