Key Concepts

• The mean (expected value) of a discrete random variable is found by multiplying each value of the random variable by its associated probability and summing the results.
• The standard deviation of a discrete random variable describes the typical distance a value is from the mean.
• As the number of trials in a probability experiment increases, the mean approaches the theoretical expected value. This is called the law of large numbers.

Glossary

expected value: expected arithmetic average when an experiment is repeated many times; also called the mean. Notations: $μ$. For a discrete random variable $(RV)$ with probability distribution function $P(x)$, the definition can also be written in the form $\mu = \sum x \cdot P(x).$

mean: a number that measures the central tendency of a distribution; a common name for mean is “average.” For a probability distribution, the mean is written as $\mu$ and it is the long-term average of many trials.

standard deviation of a probability distribution: a number that measures how far the outcomes of a statistical experiment are from the mean of the distribution $\sigma = \sqrt{\sum [(x-M)^{2} \cdot P(x)]}$.

the Law of Large Numbers: as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero