Key Concepts

Probability Distribution

A function or table that describes all the possible values of a random variable and the probabilities associated with each value.

Random Variable

A numerical variable that represents the outcome of a random phenomenon or experiment.

Discrete Random Variables

Random variables that can take on a countable number of distinct values, such as whole numbers.

Probability Distribution of a Discrete Random Variable

A list or function showing all possible values of a discrete random variable and the probability associated with each value.

Probability Distribution Function (PDF)

A function that provides the probability that a discrete random variable equals a specific value.

Cumulative Distribution Function (CDF)

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

Mean and Standard Deviation of a Discrete Random Variable

The mean (expected value) is the average outcome weighted by probabilities, and the standard deviation measures the variability or spread of the possible values around the mean.

Expected Value

The theoretical long-run average value of a random variable, calculated as the sum of all possible values weighted by their probabilities.

Bernoulli Trials

Experiments or trials that have exactly two possible outcomes (success or failure) and a constant probability of success on each trial.

Binomial Modeling

A statistical model that describes the number of successes in a fixed number of independent Bernoulli trials.

Binomial Distributions

Probability distributions that summarize the likelihood of a given number of successes in a set number of Bernoulli trials.

Continuous Random Variables

Random variables that can take on any value within a given interval or range, often associated with measurements.

Probability Distribution of a Continuous Random Variable

Described by a probability density function (PDF), representing probabilities over intervals rather than specific values.

Density Curves

Graphs that show the probability distribution of a continuous random variable; the area under the curve represents probabilities.

Normal Distribution

A continuous probability distribution that is symmetric and bell-shaped, characterized by its mean (μ) and standard deviation (σ).

Standardized Variable/Z-Score

A measure that describes a value's position relative to the mean in units of standard deviation; calculated as z=(X−μ)/σ.

Parameter

A numerical value that describes a characteristic of a population, such as the population mean (μ) or standard deviation (σ).

Statistic

A numerical value calculated from sample data, used to estimate a population parameter, such as the sample mean (x̄).

Sampling Error

The difference between a sample statistic and the corresponding population parameter, due to the randomness of sampling.

Sampling Distribution

The probability distribution of a given statistic based on a random sample; shows how the statistic varies from sample to sample.

Point Estimate

A single value calculated from sample data used to estimate a population parameter.

Interval Estimate

A range of values derived from sample data that is likely to contain the population parameter with a certain level of confidence.

Confidence Level

The probability that the interval estimate contains the population parameter; commonly used levels are 90%, 95%, and 99%.

Margin of Error

The maximum expected difference between the true population parameter and a point estimate, accounting for sampling variability.

Standard Error

The standard deviation of a sampling distribution; measures the variability of a statistic from sample to sample.

T-Distribution

A probability distribution used when estimating population parameters when the sample size is small and/or population standard deviation is unknown.

Rejection Region

The set of values for the test statistic that leads to rejecting the null hypothesis in a hypothesis test.

Non-Rejection Region

The set of values for the test statistic where the null hypothesis is not rejected.

Test Statistic

A standardized value calculated from sample data during a hypothesis test, used to decide whether to reject the null hypothesis.

Significance Level (α)

The probability threshold set by the researcher (commonly 0.05) for rejecting the null hypothesis; represents the risk of a Type I error.

P-Value

The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Type I Error

The error made when the null hypothesis is true, but is incorrectly rejected.

Type II Error

The error made when the null hypothesis is false, but is not rejected.