3 Probability Distributions

Representing uncertainty

A probability distribution describes the possible values of a random variable and the probability associated with each value or range of values. It can model outcomes such as the number of returned orders, daily sales, or delivery times. Distribution models help assess uncertainty and support statistical analyses when their assumptions are reasonable for the situation.

Discrete and continuous variables

A discrete random variable takes countable values, such as $0, 1, 2,$ and so on. Its probability mass function (PMF) gives the probability of each value, written as $P(X=k)$. The probabilities across all possible values sum to $1$.

A continuous random variable can take any value within a range, such as a time or weight. Its probability density function (PDF) describes how probability is distributed, with probabilities represented by areas under the curve. The probability of any one exact value is $0$; the probability of an interval is the area over that interval.

The cumulative distribution function (CDF) gives the probability of being at or below a specified value:

Modeling counts with the binomial distribution

Use a binomial distribution to model the number of successes in a fixed number of trials. The model requires two mutually exclusive outcomes for each trial, independent trials, and the same probability of success $p$ on every trial. For $X$, the number of successes in $n$ trials, the probability of exactly $k$ successes is

Here, $C(n,k)$ counts the different ways to get $k$ successes. The mean is $np$, and the standard deviation is $\sqrt{np(1-p)}$.

For example, suppose $10\%$ of orders are expected to be returned. For $10$ independent orders, the probability that exactly $2$ are returned is

The chance is therefore about $19.4\%$. The model is unsuitable if the return probability changes across orders or the outcomes are not independent.

Modeling measurements with the normal distribution

The normal distribution is a continuous, symmetric, bell-shaped model described by its mean $\mu$ and standard deviation $\sigma$. It is written $X\sim N(\mu,\sigma)$ when the second parameter denotes standard deviation. About $68\%$, $95\%$, and $99.7\%$ of values fall within $1$, $2$, and $3$ standard deviations of the mean, respectively.

To find probabilities, convert a value $x$ to a z-score:

The z-score indicates how many standard deviations $x$ is from the mean. A standard normal table or statistical software gives the area to the left of a z-score. Subtracting CDF values gives the probability between two values.

For example, suppose delivery time is approximately normal with a mean of $30$ minutes and a standard deviation of $5$ minutes. For a $40$-minute delivery,

The probability of a delivery taking longer than $40$ minutes is about $0.023$, or $2.3\%$.

Choosing and applying a distribution

Choose a distribution to match the quantity and process being modeled: use the binomial distribution for counts of successes in a fixed number of trials, and the normal distribution for continuous measurements that are reasonably symmetric and bell-shaped.

A model can estimate the likelihood of outcomes, such as the number of returned orders or the proportion of deliveries exceeding a service target. These estimates can help inform staffing, inventory, or quality-control decisions. Check the model's assumptions before relying on its probabilities.

In short, discrete distributions assign probability to individual outcomes, while continuous distributions use areas to represent interval probabilities. The binomial model requires fixed, independent trials with a constant success probability; the normal model describes bell-shaped continuous measurements and uses z-scores to calculate probabilities.