Probability Functions

For a function $f$ to be a valid probability function, it needs to obey these rules:

1. The integral of the function must be equal to 1, so that it is a valid event space. If the function is discrete, then the sum of the function must be equal to 1.
2. The function must be non-negative at all open sub-intervals. This is a fancy way of saying that the integral or sum over any subrange of the function cannot be negative.

Continuous valued probability distributions are called PDFs, or probability density functions. Discrete or discontinuous valued probability distributions are called PMDs, probability mass functions, or simply probability functions.

PDFs and PMFs differ in a key way. With PMFs, probabilistic events can be defined as the value of the function at some point. For example, if we have some random variable $X$ with associated probability mass function $L(X)$, then $P(X = n)$, $P(n < X < m)$ and any other coherent equation or inequality on is an evaluable event. These expressions can be treated as full probabilistic events, with all the theorems that apply. $P(X = n)$ is just the value of the function $L$ at $n$, and $P(n < X < m)$ is the integral of the function $L$ from $n$ to $m$.

By contrast, PDFs do not constitute probabilistic events when evaluated at a single value. If we have some random variable Y with associated PDF $Q(Y)$, then the event $P(Y = n)$ is not defined for any $n$. An event defined by a random variable with a continuous valued distribution has to be a range. $P(Y > n)$ is definable, as is $P(n < Y < m)$ , and they are just the integral of $Q$ from $n$ to $\infty$, and from $n$ to $m$, respectively.