October 8

In many simulations we want a value to be calculated ``randomly,'' that is chosen from some distribution of values where each particularly choice is unknowable in advance. Since we're simulating this on a computer where there has to be some rule for calculating the values, we usually use a pseudo-random sequence of values that obey our desired distribution of values, and instead of ``unknowable'' we settle for hard-to-know in advance.

Some of the distributions we simulate are

uniform:

Each value in some range has an equal chance of occurring (for example we expect each number on a single fair die to occur in 1/6 of the rolls)

normal:

Two or more independent events are summed (e.g. two dice), yielding a familiar bell-shaped curve.

exponential:

The range of values is infinite, but we determine in advance what the mean (average) value is. Streetcars may arrive every 10 minutes on average, but this includes cases where five arrive bunched together plus the cases when equipment failure suspends service for several hours.

As well as these distributions we distinguish discrete distributions (there is a finite set of possible values), and continuous distributions (infinitely many values along some segment of the real line).

We describe probability that a value, or range of values, will occur by inventing a random variable $X$, a probability density function $f_X(x)$ indicating the likelihood that $X$ will be $x$ (note the upper/lower case distinction), setting $f_X(x)$ to some value in $[0,1]$, and insisting that the sum of $f_X(x)$ for all $x$ is 1. If $X == x$ is impossible, then $f_X(x)$ is zero. If $X == x$ is certain, then $f_X(x)$ is 1. Most cases are between these extremes.