Probability distribution
A probability distribution describes every value a random variable can take and how likely each value is. The probabilities across all possible outcomes always sum to 1, whether the variable is discrete or continuous.
A probability distribution is the complete map between a random variable's possible outcomes and their probabilities. Roll a fair six-sided die, and the distribution assigns probability 1/6 to each of the values 1 through 6. Two properties hold for every valid distribution: each probability is between 0 and 1, and the probabilities sum to 1.
Distributions come in two families. A discrete distribution covers a countable set of outcomes and is described by a probability mass function — the binomial distribution (number of successes in n independent trials) and the geometric distribution (trials until the first success) are standard examples. A continuous distribution covers an unbroken range of values and is described by a probability density function, where probability corresponds to area under the curve; the normal distribution's bell curve is the most familiar case. A cumulative distribution accumulates these probabilities, giving the chance the variable is at or below each value.
Probability distributions matter because they turn uncertainty into something you can compute with. From a distribution you can find the expected value (the long-run mean), the variance and standard deviation (spread), and the probability of any event of interest — the foundations of statistical inference, quality control, and engineering reliability analysis.
The AP Statistics exam tests discrete and cumulative probability distributions, expected value, and the binomial setting in depth, while the FE Mechanical exam covers common distributions — including the normal and binomial — within its probability and statistics section. For both, know the sum-to-1 requirement, how to compute an expected value, and which named distribution fits a described scenario.
Key takeaways
- A probability distribution assigns a probability to every possible value of a random variable.
- All probabilities lie between 0 and 1 and sum to exactly 1.
- Discrete distributions use probability mass functions; continuous distributions use density functions where area equals probability.
- Expected value and standard deviation are computed directly from the distribution.
- AP Statistics and the FE Mechanical exam both test recognizing and applying common distributions like the binomial and normal.