Probability Distributions: Normal, Binomial, Poisson

A credit-card charge of $48.20, 20 signups from 200 visitors, and 10 support tickets in one hour all involve chance. They do not come from the same kind of process, so one probability model cannot answer all three questions well.

A random variable is a quantity whose value is produced by an uncertain process. A transaction amount, a count of successful signups, and a count of incoming tickets are all random variables. A probability distribution describes which values a random variable can take and how likely those values are. Unlike the descriptive distribution from the previous lesson, a probability distribution models what the process may produce next.

The first decision is not a formula. We identify what the process produces: a measured amount, a number of successes from a fixed number of attempts, or a count of events during a fixed interval.

Transaction amounts and the normal distribution

When transaction amounts vary around a typical amount, a normal distribution can model the continuous values. Continuous means the variable can take values along a scale, such as $48.20 or $48.21, rather than only whole-number counts.

A normal distribution has two parameters. Its mean, $\mu$, locates the center. Its standard deviation, $\sigma$, describes the typical distance from that center. For example, a sample of transaction amounts may have a mean near $50 and a standard deviation near $8. A transaction of $66 is then above the mean, but we need the standard deviation to judge how unusual it is.

A z-score standardizes a value by reporting its distance from the mean in standard deviation units.

For a $66 transaction in a distribution with $\mu=50$ and $\sigma=8$ ...