Probability Distribution
A probability distribution is a mathematical function that describes the likelihood of each possible outcome of a random variable. It assigns a probability to every possible value or range of values that the variable can take, with all probabilities summing to 1. Probability distributions are foundational in statistics and are used in fields ranging from insurance and finance to physics and machine learning.
Key Formula
P(X = x) = f(x), and the sum of all f(x) = 1
LaTeX: P(X = x) = f(x), \quad \sum_{x} f(x) = 1
| Symbol | Meaning | Unit |
|---|---|---|
| X | random variable | unitless |
| x | a specific value that X can take | unitless |
| f(x) | probability mass or density function evaluated at x | unitless |
Worked Example
Problem
A fair six-sided die is rolled. List the probability distribution of the outcome X.
Solution
Step 1: Identify all possible outcomes: x = 1, 2, 3, 4, 5, 6. Step 2: Since the die is fair, each outcome is equally likely. Step 3: P(X = x) = 1/6 for each value of x. Step 4: Verify: sum = 6 × (1/6) = 1. ✓
Answer
P(X = x) = 1/6 for x ∈ {1, 2, 3, 4, 5, 6}
Types of Probability Distributions
| Distribution | Variable Type | Key Parameter(s) | Common Use |
|---|---|---|---|
| Binomial | Discrete | n, p | Count of successes in n trials |
| Poisson | Discrete | λ (rate) | Events per unit time/space |
| Normal | Continuous | μ, σ | Natural and measurement data |
| Uniform | Continuous/Discrete | a, b | Equal-probability outcomes |
| Exponential | Continuous | λ | Time between events |
Interactive Tools
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Related Terms
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is a discrete probability distribution characterised by two parameters: n (number of trials) and p (probability of success on each trial). It is widely used in quality control, clinical trials, and survey analysis.
Normal Distribution
The normal distribution is a continuous, symmetric, bell-shaped probability distribution characterised by its mean (μ) and standard deviation (σ). It is the most important distribution in statistics because many natural phenomena — heights, measurement errors, test scores — follow or approximate it. The Central Limit Theorem guarantees that the mean of a large sample from any distribution is approximately normally distributed.
Expected Value
The expected value (or expectation) of a random variable is the probability-weighted average of all possible values it can take. It represents the long-run average outcome if the experiment were repeated many times. Expected value is central to decision theory, gambling, insurance, and financial risk analysis.
From Latin probabilitas (likelihood, credibility) and distributio (arrangement, apportionment). The mathematical framework was formalised by Pierre-Simon Laplace and Jacob Bernoulli in the 17th–18th centuries.