Editing Probability density function (section)

==Example==

[[File:4 continuous probability density functions.png|thumb|Examples of four continuous probability density functions.]]

Suppose bacteria of a certain species typically live 20 to 30 hours. The probability that a bacterium lives {{em|exactly}} 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.

In this example, the ratio (probability of living during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour<sup>−1</sup>). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour<sup>−1</sup>. This quantity 2 hour<sup>−1</sup> is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as (2 hour<sup>−1</sup>) ''dt''. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where ''dt'' is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour<sup>−1</sup>)×(1 nanosecond) ≈ {{val|6e-13}} (using the [[Conversion of units|unit conversion]] {{val|3.6e12}} nanoseconds = 1 hour).

There is a probability density function ''f'' with ''f''(5 hours) = 2 hour<sup>−1</sup>. The [[integral]] of ''f'' over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.