Probability Distribution Types

Why This Matters

Probability distributions are the mathematical backbone of business analytics—they're how you translate real-world uncertainty into quantifiable predictions. Whether you're forecasting demand, assessing quality control risks, or modeling customer behavior, you're being tested on your ability to select the right distribution for a given scenario. This isn't just about memorizing formulas; it's about understanding when discrete vs. continuous models apply, how parameters shape outcomes, and why certain distributions emerge from specific data-generating processes.

The exam will push you to distinguish between distributions that look similar but behave differently. A Poisson and a binomial can both count events, but they model fundamentally different situations. A normal and an exponential are both continuous, but one is symmetric and the other is skewed. Don't just memorize the formulas—know what real-world process each distribution represents and what assumptions must hold for it to be valid.

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Binary and Count-Based Discrete Distributions

These distributions model situations where you're counting discrete outcomes—successes, failures, or events. The key distinction is whether trials are independent, how many trials occur, and whether you're sampling with or without replacement.

Bernoulli Distribution

• Single trial with two outcomes—the simplest building block for all binary models, where success = 1 and failure = 0
• One parameter: $p$ (probability of success), with mean $= p$ and variance $= p(1-p)$
• Foundation for binomial and geometric distributions—understand this first, and the others follow logically

Binomial Distribution

• Counts successes in $n$ fixed, independent trials—each trial has the same probability $p$ of success
• Parameters: $n$ (trials) and $p$ (success probability); mean $= np$, variance $= np(1-p)$
• Business applications include quality control and A/B testing—use when sample size is fixed and trials are independent

Geometric Distribution

• Counts trials until the first success—models "how long until something happens" in discrete time
• Single parameter $p$ (success probability); mean $= \frac{1}{p}$, capturing the expected wait time
• Useful for customer acquisition and churn analysis—when you need to know how many attempts before conversion

Hypergeometric Distribution

• Counts successes when sampling without replacement—critical distinction from binomial, which assumes replacement
• Three parameters: $N$ (population), $K$ (successes in population), $n$ (sample size)
• Required for small populations or finite lots—use in quality inspection when you can't assume independence between draws

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Event Rate Distributions

These distributions model how often events occur over time or space. The underlying mechanism is a Poisson process—events happen randomly and independently at some average rate.

Poisson Distribution

• Counts events in a fixed interval of time, space, or other continuous measure—arrivals, defects, accidents
• Single parameter $\lambda$ (average rate); uniquely, mean $= \lambda$ and variance $= \lambda$
• Assumes events are independent and occur at constant rate—violations (clustering, seasonality) break the model

Exponential Distribution

• Models time between Poisson events—the continuous counterpart to the discrete Poisson count
• Single parameter $\lambda$ (rate); mean $= \frac{1}{\lambda}$, with the memoryless property (past waiting doesn't affect future)
• Essential for queuing theory and reliability analysis—service times, equipment failure, customer interarrival times

Gamma Distribution

• Generalizes exponential to model time until the $k$-th event—when you need more than one event to occur
• Two parameters: shape $k$ and scale $\theta$ (or rate $\beta = \frac{1}{\theta}$); mean $= k\theta$
• Used in insurance claims and project duration modeling—when waiting for multiple independent stages to complete

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Continuous Distributions for Measurement Data

These distributions model variables that can take any value within a range. They're defined by probability density functions, where area under the curve—not individual points—represents probability.

Normal Distribution

• Symmetric, bell-shaped curve defined by mean $\mu$ (center) and standard deviation $\sigma$ (spread)
• Central Limit Theorem makes this critical—sample means approach normal regardless of the underlying distribution
• Standard normal ($Z$-distribution) has $\mu = 0$, $\sigma = 1$—used for hypothesis testing and confidence intervals

Uniform Distribution

• All outcomes equally likely between minimum $a$ and maximum $b$—the "I have no idea" distribution
• Mean $= \frac{a+b}{2}$, variance $= \frac{(b-a)^2}{12}$—flat probability density across the entire range
• Used in simulation and as a prior when no information exists—also models rounding errors and random number generation

Beta Distribution

• Defined only on $[0, 1]$—perfect for modeling proportions, probabilities, and percentages
• Two shape parameters $\alpha$ and $\beta$ create flexible shapes: symmetric, skewed, U-shaped, or uniform
• Bayesian statistics workhorse—used as a prior for unknown probabilities and in PERT project estimation

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Quick Reference Table

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Self-Check Questions

1. A quality inspector examines 15 items from a shipment of 100, where 8 are known to be defective. Which distribution models the number of defectives found—binomial or hypergeometric? Why?

2. Compare the Poisson and binomial distributions: under what conditions does binomial approximate Poisson, and what real-world scenario would make this approximation useful?

3. If customer arrivals follow a Poisson distribution with $\lambda = 12$ per hour, what distribution describes the time between consecutive arrivals, and what is its mean?

4. You need to model the probability that a new product captures between 15% and 25% of market share. Which distribution is most appropriate, and what makes it suitable for this scenario?

5. Explain why the Central Limit Theorem makes the normal distribution essential for business statistics, even when the underlying data isn't normally distributed.