Fundamentals of Probability

Bernoulli (or Binary) Random Variable

A random variable that takes on only two values (usually 0 and 1)

Binomial Distribution

The probability distribution of the number of successes in a fixed number of independent Bernoulli trials

Chi-Square Distribution

A distribution obtained by summing the squares of independent standard normal random variables

Conditional Distribution

The probability distribution of a random variable given that another variable takes on a specific value

Conditional Expectation

The expected value of a random variable calculated given the occurrence of some specific event

Continuous Random Variable

A variable that can take on any value within a specific range or interval

Correlation Coefficient

A standardized measure of the linear relationship between two variables ranging from -1 to 1

Covariance

A measure of the joint variability of two random variables

Cumulative Distribution Function (cdf)

A function giving the probability that a random variable is less than or equal to a certain value

Degrees of Freedom

The number of independent values or quantities that can be assigned to a statistical distribution

Discrete Random Variable

A variable that can only take on a finite or countable number of distinct values

Expected Value

The long-run average value of a random variable

Experiment

Any process that generates a set of data or outcomes

F Distribution

A continuous probability distribution that arises frequently as the null distribution of a test statistic

Independent Random Variables

Variables where the occurrence of one does not affect the probability of the other

Joint Distribution

The probability distribution that simultaneously describes two or more random variables

Kurtosis

A measure of the "tailedness" of the probability distribution of a real-valued random variable

Law of Iterated Expectations

A rule stating that the unconditional expectation is equal to the expectation of the conditional expectation

Median

The value separating the higher half from the lower half of a data sample

Normal Distribution

A bell-shaped symmetric continuous distribution defined by its mean and variance

Pairwise Uncorrelated Random Variables

A set of variables where every possible pair has a correlation of zero

Probability Density Function (pdf)

A function that describes the relative likelihood for a continuous random variable to take on a given value

Random Variable

A variable whose values are determined by the outcomes of a random phenomenon

Skewness

A measure of the asymmetry of the probability distribution of a real-valued random variable

Standard Deviation

The square root of the variance representing the average distance from the mean

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1

Standardized Random Variable

A variable transformed to have a mean of 0 and a variance of 1

Symmetric Distribution

A distribution where the left and right sides are mirror images of each other

t Distribution

A distribution used for estimating population parameters when the sample size is small

Uncorrelated Random Variables

Variables with a correlation coefficient of zero

Variance

The expectation of the squared deviation of a random variable from its mean