| Term | Definition |
|---|---|
| binomial distribution | A probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. |
| binomial probability function | The formula P(X=x)=C(n,x)p^x(1-p)^(n-x) that calculates the probability of exactly x successes in n independent trials with probability of success p. |
| binomial random variable | A random variable that counts the number of successes in a fixed number of repeated independent trials, where each trial has two possible outcomes. |
| independent trials | Repeated experiments or observations where the outcome of one trial does not affect the outcome of any other trial. |
| number of failures | The count of unfavorable outcomes in a sample, denoted as n(1-p̂), used to verify the normality condition. |
| number of successes | The count of favorable outcomes in a sample, denoted as np̂, used to verify the normality condition. |
| probability distribution | A function that describes the likelihood of all possible values of a random variable. |
| probability of success | The constant probability p that an individual trial results in a success in a binomial experiment. |
| random number generator | A tool or method used to randomly select items from a population for inclusion in a simple random sample. |
| simulation | A method of modeling random events so that simulated outcomes closely match real-world outcomes, used to estimate probabilities. |
| Term | Definition |
|---|---|
| patterns in data | Observable regularities or trends that appear in a dataset, which may or may not indicate non-random behavior. |
| variation | Differences in data that occur by chance due to the random nature of sampling, rather than from systematic causes. |
| Term | Definition |
|---|---|
| binomial distribution | A probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| parameter | A numerical summary that describes a characteristic of an entire population. |
| probability | The likelihood or chance that a particular outcome or event will occur, expressed as a value between 0 and 1. |
| random variable | A variable whose value is determined by the outcome of a random phenomenon and can take on different numerical values with associated probabilities. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| Term | Definition |
|---|---|
| geometric distribution | A probability distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, each with the same probability of success. |
| geometric probability function | The formula P(X=x)=(1-p)^(x-1)p that calculates the probability that the first success occurs on trial x. |
| geometric random variable | A random variable that represents the number of the trial on which the first success occurs in a sequence of independent trials. |
| independent trials | Repeated experiments or observations where the outcome of one trial does not affect the outcome of any other trial. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| number of failures | The count of unfavorable outcomes in a sample, denoted as n(1-p̂), used to verify the normality condition. |
| number of successes | The count of favorable outcomes in a sample, denoted as np̂, used to verify the normality condition. |
| parameter | A numerical summary that describes a characteristic of an entire population. |
| probability | The likelihood or chance that a particular outcome or event will occur, expressed as a value between 0 and 1. |
| probability of success | The constant probability p that an individual trial results in a success in a binomial experiment. |
| random variable | A variable whose value is determined by the outcome of a random phenomenon and can take on different numerical values with associated probabilities. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| Term | Definition |
|---|---|
| event | A collection of one or more outcomes from a random process. |
| law of large numbers | The principle that simulated or empirical probabilities tend to get closer to the true probability as the number of trials increases. |
| outcome | The result of a single trial of a random process. |
| random process | A process that generates results determined by chance, where the outcome cannot be predicted with certainty in advance. |
| relative frequency | The proportion of observations in a category, expressed as a decimal, fraction, or percentage of the total. |
| simulation | A method of modeling random events so that simulated outcomes closely match real-world outcomes, used to estimate probabilities. |
| Term | Definition |
|---|---|
| complement of an event | The set of all outcomes in the sample space that are not in event E, denoted E' or E^C, representing 'not E'. |
| equally likely | A condition where all outcomes in a sample space have the same probability of occurring. |
| event | A collection of one or more outcomes from a random process. |
| long run | A large number of repetitions of a probability experiment where the relative frequency of an event approaches its true probability. |
| outcome | The result of a single trial of a random process. |
| probability | The likelihood or chance that a particular outcome or event will occur, expressed as a value between 0 and 1. |
| random process | A process that generates results determined by chance, where the outcome cannot be predicted with certainty in advance. |
| relative frequency | The proportion of observations in a category, expressed as a decimal, fraction, or percentage of the total. |
| sample space | The set of all possible non-overlapping outcomes of a random process. |
| Term | Definition |
|---|---|
| intersection | The set of outcomes that belong to both event A and event B, denoted A ∩ B. |
| joint probability | The probability that two events A and B both occur, denoted P(A ∩ B). |
| mutually exclusive | Two events that cannot occur at the same time; events with no outcomes in common. |
| Term | Definition |
|---|---|
| conditional probability | The probability that one event will occur given that another event has already occurred, denoted P(A | B). |
| joint probability | The probability that two events A and B both occur, denoted P(A ∩ B). |
| multiplication rule | A probability rule stating that P(A ∩ B) = P(A) · P(B | A), used to find the probability that two events both occur. |
| Term | Definition |
|---|---|
| addition rule | A probability rule stating that P(A ∪ B) = P(A) + P(B) - P(A ∩ B), used to find the probability of the union of two events. |
| conditional probability | The probability that one event will occur given that another event has already occurred, denoted P(A | B). |
| independent events | Events A and B are independent if knowing whether event A has occurred does not change the probability that event B will occur. |
| intersection | The set of outcomes that belong to both event A and event B, denoted A ∩ B. |
| union of events | The event that either event A or event B or both will occur, denoted P(A ∪ B). |
| Term | Definition |
|---|---|
| center | A measure indicating the middle or typical value of a distribution. |
| cumulative probability distribution | A representation (as a table or function) showing the probability that a random variable is less than or equal to each of its possible values. |
| discrete random variable | A random variable that takes on a countable number of distinct values, often representing counts or categorical outcomes. |
| population | The entire group of individuals or items from which a sample is drawn and about which conclusions are to be made. |
| probability distribution | A function that describes the likelihood of all possible values of a random variable. |
| random process | A process that generates results determined by chance, where the outcome cannot be predicted with certainty in advance. |
| random variable | A variable whose value is determined by the outcome of a random phenomenon and can take on different numerical values with associated probabilities. |
| shape | The overall form or pattern of a distribution, including characteristics like skewness and modality. |
| spread | A measure of how dispersed or variable the outcomes of a probability distribution are, such as range, variance, or standard deviation. |
| Term | Definition |
|---|---|
| discrete random variable | A random variable that takes on a countable number of distinct values, often representing counts or categorical outcomes. |
| expected value | The long-run average outcome of a random variable, equivalent to the mean of a discrete random variable. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| parameter | A numerical summary that describes a characteristic of an entire population. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| Term | Definition |
|---|---|
| independent random variables | Random variables where knowing the value or probability distribution of one does not change the probability distribution of the other. |
| linear combinations | Expressions of the form aX + bY where X and Y are random variables and a and b are real number coefficients. |
| linear transformations | Changes to a random variable of the form Y = a + bX, where a and b are constants that shift and scale the distribution. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| probability distribution | A function that describes the likelihood of all possible values of a random variable. |
| random variable | A variable whose value is determined by the outcome of a random phenomenon and can take on different numerical values with associated probabilities. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| variance | A measure of the spread or dispersion of a probability distribution, denoted as σ², indicating how far values typically deviate from the mean. |