12 Discrete Probability Distributions
Chapters 9 and 10 dealt with single events: the probability a customer is satisfied, the probability an item is defective. A probability distribution goes further, it describes the probability of every possible outcome of a random process at once, turning single-event probability into a complete picture of what could happen. This chapter covers discrete distributions, where the outcomes are countable (0, 1, 2, 3 defective items, never 1.5), starting with the general idea of a random variable and its expected value, then two of the most widely used discrete distributions in business analytics: the Binomial and the Poisson.
12.1 Random Variables and Discrete Distributions
A random variable assigns a number to the outcome of a random process. It is discrete if it can only take a countable set of values (the number of defective items in a batch, the number of customer complaints in a day), and continuous if it can take any value in a range (a transaction amount, a person’s height), the subject of the next chapter.
A discrete probability distribution lists every possible value of a discrete random variable \(X\) alongside its probability \(P(x)\), and must satisfy two rules:
\[0 \leq P(x) \leq 1 \text{ for every } x \qquad \text{and} \qquad \sum P(x) = 1\]
The expected value (mean) of the distribution is a weighted average of every possible outcome, weighted by its own probability:
\[E(X) = \mu = \sum x \cdot P(x)\]
The variance measures the spread of the distribution around that expected value:
\[Var(X) = \sigma^2 = \sum (x - \mu)^2 \cdot P(x)\]
\(E(X)\) is exactly the same idea as the weighted mean from Chapter 5, just applied to every possible future outcome of a random process instead of to observed data already collected. It answers “what would the long-run average look like if this process repeated many times?”
12.2 The Binomial Distribution
The Binomial distribution applies whenever a process consists of a fixed number of independent trials, each with exactly two possible outcomes (“success” or “failure”), and the same probability of success on every trial. Four conditions must all hold:
- A fixed number of trials, \(n\).
- Each trial has only two outcomes (success/failure).
- The probability of success, \(p\), is constant across trials.
- Trials are independent of one another.
\[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \qquad \text{where } \binom{n}{x} = \frac{n!}{x!(n-x)!}\]
\[\text{Mean: } \mu = np \qquad \text{Variance: } \sigma^2 = np(1-p)\]
Example
A manufacturer knows that 5% of items coming off a production line are defective (\(p = 0.05\)), independently of one another. A quality inspector randomly samples \(n = 10\) items.
\[P(X=2) = \binom{10}{2}(0.05)^2(0.95)^8 = 45 \times 0.0025 \times 0.6634 \approx 0.0746\]
There is about a 7.5% chance that exactly 2 of the 10 sampled items are defective. The probability of finding at most 1 defective item is \(P(X=0) + P(X=1) \approx 0.5987 + 0.3151 = 0.9139\), roughly 91.4%. The expected number of defectives per sample of 10 is \(\mu = np = 10 \times 0.05 = 0.5\), with variance \(\sigma^2 = 10 \times 0.05 \times 0.95 = 0.475\).
12.3 The Poisson Distribution
The Poisson distribution applies to counts of events happening independently, at a known constant average rate, within a fixed interval of time, space, or volume, such as complaints per day, website visits per minute, or flaws per meter of fabric. Unlike the Binomial, there is no fixed “number of trials,” only an average rate \(\lambda\) (lambda).
\[P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}\]
\[\text{Mean: } \mu = \lambda \qquad \text{Variance: } \sigma^2 = \lambda\]
A distinctive feature of the Poisson distribution is that its mean and variance are always equal.
Example
A customer support center receives an average of \(\lambda = 4\) complaints per day.
\[P(X=6) = \frac{4^6 e^{-4}}{6!} \approx 0.1042\]
There’s about a 10.4% chance of exactly 6 complaints on a given day. The probability of a complaint-free day is \(P(X=0) = e^{-4} \approx 0.0183\), only about 1.8%, and the probability of 2 or fewer complaints is \(P(X \leq 2) \approx 0.2381\), about 23.8%. Both the mean and variance of daily complaints are 4.
12.4 Choosing Between Binomial and Poisson
- Binomial: Use when there is a known, fixed number of trials \(n\), each with the same success probability \(p\), such as sampling a fixed batch of items for defects.
- Poisson: Use when counting how many times something happens over a continuous interval with no natural “number of trials,” such as complaints per day or website errors per hour, only a known average rate \(\lambda\).
The Poisson distribution can also be used as a convenient approximation to the Binomial when \(n\) is large and \(p\) is small, exactly the shape of the quality-inspection example above, which is why both distributions so often appear together in the same business problem: rare, independent events counted over many opportunities.
Looking Ahead
Both distributions in this chapter describe countable outcomes. The next chapter turns to the Normal distribution, which describes continuous data instead, transaction amounts, exam scores, measurements, and it is the single most important distribution in all of statistics, forming the foundation for the sampling, estimation, and hypothesis-testing chapters that follow.
Summary
| Concept | Description |
|---|---|
| Core Idea | |
| Random Variable | A variable that assigns a numeric value to the outcome of a random process |
| Discrete Random Variable | A random variable that takes only a countable set of values |
| Discrete Probability Distribution | A list of every possible value of a discrete random variable with its probability, summing to 1 |
| Expected Value | The probability-weighted average of every possible outcome, the mean of the distribution |
| Variance of a Discrete Distribution | The probability-weighted average squared distance from the expected value |
| Binomial | |
| Binomial Distribution | Models the count of successes in a fixed number of independent trials with constant success probability |
| Binomial Conditions | Fixed number of trials, two outcomes per trial, constant probability of success, independent trials |
| Binomial Mean and Variance | Mean equals n times p; variance equals n times p times one minus p |
| Poisson | |
| Poisson Distribution | Models the count of independent events occurring at a known average rate within a fixed interval |
| Poisson Mean and Variance | Mean equals variance equals lambda, the average rate |
| Choosing | |
| Choosing Binomial vs. Poisson | Binomial needs a fixed number of trials; Poisson needs only a known average rate over an interval |