Common Probability Distributions

Why This Matters

Probability distributions are the foundation of everything you'll do in biostatistics—from designing clinical trials to interpreting p-values to modeling disease outbreaks. When you're tested on this material, you're not just being asked to recall formulas. You're being evaluated on whether you understand when to apply each distribution, what assumptions it requires, and how distributions relate to each other. The exam will present scenarios and expect you to identify the appropriate distribution based on the data type and underlying process.

Think of distributions as tools in a toolkit: the normal distribution handles continuous measurements, the binomial counts successes in fixed trials, and the Poisson tracks rare events over time. Each distribution encodes specific assumptions about how data behave—independence, fixed trials, constant rates, symmetry. Don't just memorize parameters—know what real-world process each distribution models and when one distribution approximates another.

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Continuous Distributions for Measurement Data

These distributions model variables that can take any value within a range. They're essential for analyzing measurements like blood pressure, weight, reaction times, and biomarker concentrations.

Normal (Gaussian) Distribution

• Bell-shaped and symmetric around the mean ($\mu$)—the most important distribution in statistics because of its mathematical properties and real-world prevalence
• Empirical Rule (68-95-99.7): approximately 68% of data falls within $\pm 1\sigma$, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$ of the mean
• Central Limit Theorem guarantees that sample means approach normality as $n$ increases, even when the underlying population isn't normal

Student's t-Distribution

• Heavier tails than the normal distribution—accounts for extra uncertainty when estimating population parameters from small samples
• Degrees of freedom (df) control the shape; as $df \to \infty$, the t-distribution converges to the standard normal
• Primary use: hypothesis tests and confidence intervals for means when $\sigma$ is unknown and $n < 30$

Uniform Distribution

• All outcomes equally likely between minimum ($a$) and maximum ($b$) values—the "no information" distribution
• Mean is $(a + b)/2$ and variance is $(b - a)^2/12$ for the continuous case
• Simulation workhorse: random number generators produce uniform variates that are transformed into other distributions

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Discrete Distributions for Counting Events

These distributions model outcomes you can count—number of successes, number of occurrences, binary outcomes. They're fundamental for categorical data analysis and clinical trial design.

Bernoulli Distribution

• Single trial with two outcomes: success (1) with probability $p$ or failure (0) with probability $1-p$
• Mean is $p$ and variance is $p(1-p)$—maximum variance occurs when $p = 0.5$
• Building block for the binomial distribution; understanding Bernoulli trials is essential for grasping more complex counting distributions

Binomial Distribution

• Counts successes in $n$ independent Bernoulli trials—think drug response rates, disease prevalence in samples, or treatment outcomes
• Parameters: $n$ (number of trials) and $p$ (probability of success); mean is $np$, variance is $np(1-p)$
• Normal approximation works well when $np \geq 10$ and $n(1-p) \geq 10$, making large-sample calculations tractable

Poisson Distribution

• Models count of rare events in a fixed interval of time or space, given average rate $\lambda$
• Key property: mean and variance are both equal to $\lambda$—if observed variance greatly exceeds the mean, Poisson assumptions may be violated
• Applications: hospital admissions per day, mutations per genome region, adverse events in clinical trials

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Distributions for Time-to-Event and Waiting Times

These continuous distributions model how long until something happens—critical for survival analysis, reliability studies, and pharmacokinetics.

Exponential Distribution

• Models time between events in a Poisson process, with rate parameter $\lambda$ (or equivalently, mean $1/\lambda$)
• Memoryless property: $P(T > s + t \mid T > s) = P(T > t)$—the system doesn't "age," making future predictions independent of elapsed time
• Survival analysis foundation: models time to death, equipment failure, or disease recurrence when hazard rate is constant

Gamma Distribution

• Generalizes the exponential—models the waiting time for $k$ events in a Poisson process (shape parameter $k$, scale parameter $\theta$)
• Flexible shape: can be right-skewed (small $k$) or nearly symmetric (large $k$); when $k = 1$, reduces to exponential
• Applications: total hospital length of stay, aggregate waiting times, and as a prior distribution in Bayesian analysis

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Distributions for Proportions and Hypothesis Testing

These distributions arise in specific statistical procedures—testing hypotheses, estimating proportions, and Bayesian inference.

Chi-Square Distribution

• Sum of squared standard normal variables—arises naturally when estimating variance or testing categorical data
• Degrees of freedom (df) determine shape; distribution is right-skewed but approaches normality as $df$ increases
• Primary uses: goodness-of-fit tests, tests of independence in contingency tables, and confidence intervals for variance

Beta Distribution

• Defined on the interval [0, 1]—perfect for modeling probabilities, proportions, and rates
• Shape parameters $\alpha$ and $\beta$ control the distribution: symmetric when $\alpha = \beta$, skewed otherwise; uniform when $\alpha = \beta = 1$
• Bayesian workhorse: serves as the conjugate prior for binomial data, making posterior calculations elegant

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Quick Reference Table

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Self-Check Questions

1. A researcher is counting the number of patients who respond to a new drug out of 50 treated. Which distribution models this outcome, and what parameters define it?

2. Compare and contrast the Poisson and exponential distributions. How are they mathematically related, and when would you use each?

3. Why does the t-distribution have heavier tails than the normal distribution? Under what conditions do they become equivalent?

4. You're modeling the proportion of time a patient spends in remission (a value between 0 and 1). Which distribution is most appropriate, and why?

5. An FRQ presents hospital emergency room data showing the mean number of arrivals per hour equals 4, but the variance equals 12. Should you use a Poisson model? Explain your reasoning using the distribution's key property.