Types of Probability Distributions

Why This Matters

Probability distributions are the mathematical models that let statisticians predict outcomes, quantify uncertainty, and make inferences about populations. On the AP Statistics exam, you're being tested on your ability to recognize which distribution applies to a given scenario, understand the conditions required for each model, and calculate probabilities using their specific parameters. These distributions connect directly to Units 4, 5, and beyond—from basic probability calculations to sampling distributions to inference procedures like confidence intervals and hypothesis tests.

Don't just memorize formulas and shapes. Know why each distribution exists: What real-world process does it model? What conditions must be met? How does it connect to the Central Limit Theorem or chi-square tests? When you understand the underlying mechanism, you can tackle any FRQ scenario the exam throws at you—whether it's identifying the right distribution, checking conditions, or interpreting results in context.

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Discrete Distributions: Counting Successes and Events

These distributions model situations where you're counting discrete outcomes—how many successes occur, how many trials until success, or how many events happen in a given interval. Each has specific conditions that determine when it's the appropriate model.

Bernoulli Distribution

• Models a single trial with exactly two outcomes—success (1) or failure (0), making it the simplest probability distribution
• Single parameter $p$ represents the probability of success; the probability of failure is $1-p$
• Foundation for the binomial distribution—a binomial is simply the sum of $n$ independent Bernoulli trials

Binomial Distribution

• Counts successes in $n$ fixed, independent trials—each trial must have the same probability of success $p$ (think BINS: Binary, Independent, Number fixed, Same probability)
• Parameters $n$ and $p$ fully define the distribution; mean is $\mu = np$ and standard deviation is $\sigma = \sqrt{np(1-p)}$
• Large counts condition ($np \geq 10$ and $n(1-p) \geq 10$) allows normal approximation—critical for confidence intervals on proportions

Poisson Distribution

• Models rare events in a fixed interval—counts occurrences when events happen independently at a constant average rate $\lambda$
• Mean equals variance (both equal $\lambda$)—a unique property that helps identify Poisson scenarios
• Approximates binomial when $n$ is large and $p$ is small—useful for rare event modeling like defects or accidents

Geometric Distribution

• Counts trials until first success—models how long you wait for something to happen in repeated independent Bernoulli trials
• Single parameter $p$ (probability of success); mean number of trials is $\mu = \frac{1}{p}$
• Memoryless property—the probability of success on the next trial doesn't depend on how many failures came before (each trial is a fresh start)

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Continuous Distributions: Modeling Measurements

Continuous distributions model variables that can take any value within an interval—time, height, test scores, or any measurement on a continuous scale. Probability is found as area under the density curve, not at individual points.

Uniform Distribution

• All outcomes equally likely within a defined range from $a$ to $b$—the simplest continuous distribution
• Constant probability density of $\frac{1}{b-a}$ across the entire interval; mean is $\frac{a+b}{2}$
• Can be discrete or continuous—discrete uniform applies to equally likely categorical outcomes (like rolling a fair die)

Normal Distribution

• Symmetric, bell-shaped curve defined by mean $\mu$ (center) and standard deviation $\sigma$ (spread)
• Empirical Rule (68-95-99.7)—approximately 68% of data falls within $1\sigma$ of the mean, 95% within $2\sigma$, 99.7% within $3\sigma$
• Central to inference procedures—the Central Limit Theorem guarantees sampling distributions approach normality, enabling z-based confidence intervals and hypothesis tests

Exponential Distribution

• Models waiting time between events—how long until the next occurrence when events happen at rate $\lambda$
• Memoryless property—the probability of waiting another $t$ minutes is the same regardless of how long you've already waited
• Connected to Poisson—if events occur according to a Poisson process with rate $\lambda$, the time between events follows an exponential distribution

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Sampling and Inference Distributions

These distributions arise specifically in statistical inference—they describe how test statistics behave under the null hypothesis or how estimators vary across samples. Understanding their shapes and parameters is essential for hypothesis testing.

Student's t-Distribution

• Heavier tails than normal—accounts for extra uncertainty when estimating population standard deviation from sample data
• Degrees of freedom (df) control the shape; as df increases, the t-distribution approaches the standard normal
• Used when $\sigma$ is unknown—essential for confidence intervals and hypothesis tests about means with small samples or unknown population SD

Chi-Square Distribution

• Always positive and right-skewed—models the distribution of squared standardized values, so it can't be negative
• Degrees of freedom determine shape; larger df means less skew and more symmetric appearance
• Powers goodness-of-fit and independence tests—compares observed counts to expected counts using $\sum \frac{(O-E)^2}{E}$

F-Distribution

• Ratio of two chi-square distributions—used to compare variances or test whether multiple group means differ
• Two degrees of freedom parameters (numerator and denominator)—order matters for calculating critical values
• Always positive and right-skewed—approaches normality as both df increase; central to ANOVA procedures

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Quick Reference Table

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Self-Check Questions

1. A quality control inspector examines 100 items and records how many are defective. Which distribution models this scenario, and what conditions must be verified?

2. Compare the geometric and exponential distributions: What do they have in common, and how do their applications differ?

3. Why does the AP Statistics curriculum emphasize the normal distribution so heavily? Connect your answer to the Central Limit Theorem and inference procedures.

4. An FRQ presents a chi-square test for independence. Explain why the chi-square distribution (rather than the normal or t) is the appropriate sampling distribution for the test statistic.

5. Both the Poisson and binomial distributions count discrete events. Under what conditions can you use the Poisson as an approximation for the binomial, and why might you want to?