Cumulative Probability Functions

It is useful in many project situations to think of the accumulating probability of an event happening. For instance, it might be useful to convey to the project sponsor that "...there is a 0.6 probability that the schedule will be 10 weeks or shorter." Since the maximum cumulative probability is 1, at some point the project manager can declare "...there is certainty, with probability 1, that the schedule will be shorter than x weeks."

We already have the function that will give us this information; we need only apply it. If we sum up the probability functions of X over a continuous range of values, a_i, then we have what we want: ∑ all f_i(X | a_i) = 1, for i = "m" to "n" accumulates the probabilities of values between the limits of "m" and "n".

Table 2-2 provides an example of how a cumulative probability function works for a discrete random variable.

Table 2-2: Cumulative Discrete Probability Function
A	B	C
Outcome of Random Variable D_i for an Activity Duration	Probability Density of Outcome D_i	Cumulative Probability of Outcome D_i
3 days	0.1	0.1
5 days	0.3	0.4
7 days	0.4	0.8
10 days	0.15	0.95
20 days	0.05	1.0

D_i is an outcome of an event described by the random variable D for task duration.
The probability of a single-valued outcome is given in column B; the accumulating probability that the duration will be equal to or less than the outcome in column A is given in column C.

Of course, for a continuous random variable, it is pretty much the same. We integrate from one limit of value to another to find the probability of the value of X hitting in the range between the limits of integration. For our purposes, integration is nothing more than summation with arbitrarily small separation between values.

^[6]Italicized bold capital letters will be used for random variables.

^[7]The probability function is often called the "probability density function." This name helps distinguish it from the cumulative probability function and also fits with the idea that the probability function really is a density, giving probability per value.

^[8]"dX" is a notation used to mean a small, but not zero, value. Readers familiar with introductory integral calculus will recognize this convention.