Probability Distribution: Definition, Types, and Uses in Investing

A probability distribution describes the likelihood of every possible outcome for a random variable. It assigns probabilities to outcomes (discrete case) or describes their density over a range of values (continuous case). Distributions are used across science, engineering and finance to summarize uncertainty and to make probabilistic forecasts.

Key takeaways

• A probability distribution shows which outcomes are most and least likely.
• Distributions are categorized as discrete (countable outcomes) or continuous (any value within a range).
• Common distributions include binomial, normal, Poisson and exponential.
• In finance, distributions help estimate likely returns, tail risks and metrics such as Value at Risk (VaR).
• The central limit theorem explains why sample means often look normally distributed for large samples.

How probability distributions work

• Probability density function (PDF): for continuous variables, the PDF describes relative likelihood across values. Area under the PDF over an interval gives the probability of falling in that interval.
• Probability mass function (PMF): for discrete variables, the PMF gives the probability of each specific outcome.
• Cumulative distribution function (CDF): gives the probability a variable is less than or equal to a given value; it runs from 0 to 1.
• Sampling considerations: empirical distributions based on limited historical data can suffer from sampling error; larger samples reduce that error.

Discrete vs. continuous distributions

• Discrete distributions: outcomes are countable (e.g., number of heads in 10 coin flips, number of customers arriving). The PMF lists probabilities for each possible value.
• Continuous distributions: outcomes form a continuum (e.g., heights, exact time to complete a task). Probabilities are expressed via areas under the PDF; the probability of any single exact value is effectively zero.

Common probability distributions

• Binomial: discrete. Models the number of successes in a fixed number of independent trials with the same success probability (e.g., number of made free throws out of N attempts).
• Bernoulli: special case of the binomial with a single trial (success/failure).
• Normal (Gaussian): continuous and symmetric; fully described by mean and standard deviation. Many statistical methods rely on normality, and the empirical rule states ~68% of data lie within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD.
• Poisson: discrete. Models counts of independent events occurring at a constant average rate over an interval (e.g., arrivals per hour) and is useful for rare-event counts.
• Exponential: continuous. Models time between independent events in a Poisson process.
• Uniform: outcomes within an interval are equally likely.

Example: two six-sided dice

Rolling two standard dice produces sums from 2 to 12. The probabilities are not uniform because some sums have more combinations (7 is most likely with six combinations; 2 and 12 are least likely with one combination each). This illustrates how a distribution summarizes outcome likelihoods.

What makes a distribution valid?

A probability distribution is valid if:
1. Every individual probability (or density) is ≥ 0.
2. The total probability over all possible outcomes equals 1 (for continuous distributions, the integral of the PDF over its domain equals 1).

Uses in investing

• Estimating returns: distributions of historical returns help project likely future outcomes, though past returns are not guarantees.
• Risk measurement: distributions quantify downside risk and tail events; Value at Risk (VaR) is a common metric that gives the minimum loss at a specified probability over a time horizon.
• Modeling asset prices: stock prices are often modeled as lognormal (prices bounded below by zero with potentially unbounded upside), and empirical return distributions frequently show heavier tails (higher kurtosis) than the normal distribution—meaning large gains and losses occur more often than a normal model would predict.
• Caution: relying solely on historical distributions or a single risk metric (like VaR) can understate true risk, particularly for rare but severe events.

Central Limit Theorem and Law of Large Numbers

• Central Limit Theorem (CLT): the distribution of the sum (or average) of many independent, identically distributed random variables tends toward a normal distribution, regardless of the original variable’s distribution, provided the sample size is large enough. This underpins many inferential methods.
• Law of Large Numbers: as the number of independent trials increases, the sample average converges to the expected (true) value. More observations generally improve the accuracy of estimates.

Common questions

• What is the difference between probability and odds?
  Probability is the chance an event occurs (0 to 1). Odds compare the probability of occurrence to non-occurrence (e.g., probability 0.25 → odds 1:3).
• Which distributions are most used?
  Frequently used distributions include normal, binomial, Poisson, Bernoulli, exponential and uniform.
• Why is the normal distribution used so often?
  Its mathematical properties, the CLT, and tractability make it convenient; however, real-world data—especially financial returns—can deviate from normality (skewness, heavy tails).

Bottom line

Probability distributions are fundamental tools for describing uncertainty. Choosing the appropriate distribution and understanding its assumptions and limitations are essential for accurate modeling, particularly in finance where tail risks and non-normal behavior matter.