Continuous distributions

Things are slightly less intuitive when our event can be selected from a segment of points on the real line. For example, I usually tell anyone who cares that I'm 166 centimeters tall. However, I'm sure that a precise enough measure would determine that I'm not exactly 166.000000...cm tall. Adult human heights are distributed over some interval between 1 meter and 3 meters (to be generous), but the chance that some particular exact height is achieved is basically zero.

So we change our point of view and talk about the probability that our value falls within some particular range. We define $f_X(x)$ so that its values are always non-negative and its integral from $a$ to $b$ represents the probability that $X$ takes a value in $[a,b]$:

$$\int_a^b f_X(t) dt =$$   probability that $X$ is in $[a,b]$$$.$$

From the continuous point of view, the uniform distribution over the interval $[a,b]$ needs an $f_X$ whose integral is 1 and whose value for any two equally-wide subintervals of $[a,b]$ is the same:

$$f_X(x) = \begin{cases} 1/(b-a) & x \in [a,b] \\ 0 & \text{otherwise} \end{cases}$$

Suppose you want a distribution that models the distribution of times between events that happen at rate $r$ (reciprocal of mean time between events), but you have no upper limit on the maximum possible time. A distribution that works is the exponential distribution:

$$f_X(t) = \begin{cases} r e^{-rt} & t \geq 0, \\ 0 & \text{otherwise} \end{cases}.$$