Entire Exam 2 notes

Probability

The likelihood of an event occurring based on repeated experiments under identical conditions.

Probability Notation

Denoted as P(E), where 'E' is any event.

Probabilities Range

Probability values must fall between 0 (impossible event) and 1 (certain event).

Mutually Exclusive Events

Events that cannot happen simultaneously.

Independent Events

The occurrence of one event does not affect the occurrence of another event.

Sample Space

The set of all possible outcomes of an experiment.

Relative Frequency Assessment

An empirical way to estimate probabilities based on past occurrences.

Subjective Probability Assessment

A personal estimation of an event's chance occurring, often without statistical backing.

Conditional Probability

Probability of an event given another event has occurred.

Binomial Distribution

Probability distribution applicable for a fixed number of independent trials with two outcomes.

Expected Value (Mean)

The average value calculated by E(X) = Σ (x_i * P(x_i)).

Variance

A measure of how spread out the values of a random variable are.

Standard Deviation

A measure of the amount of variation or dispersion in a set of values.

Z-Score

A statistic that tells you how many standard deviations a data point is from the mean.

Exponential Distribution

Continuous probability distribution used to model time until an event occurs.

Hypergeometric Distribution

Used for modeling scenarios where the selection is done without replacement.

Cumulative Probability

The probability that the random variable takes a value less than or equal to a certain value.

Normal Distribution

A probability distribution that is symmetric about the mean, representing data that clusters around an average.

Uniform Distribution

Continuous probability distribution where all outcomes are equally likely within a specified range.

Poisson Distribution

A probability distribution used for modeling count data over a specified period of time.

Probability Notation

Notated as P(E), where E is any event, representing the likelihood of that event occurring.

Mutually Exclusive Events

Events that cannot happen simultaneously; the occurrence of one excludes the other.

Independent Events

The occurrence of one event does not affect the occurrence of another event.

Addition Rule of Probability

P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2); used to find the probability of at least one of two events occurring.

Expected Value

E(X) = Σ (Xi * P(Xi)); the average outcome of a random variable X based on its probability distribution.

Variance

A measure of how much values differ from the expected value; for binomial variance, it is a²(X) = npq.

Combinations Formula

C(n, x) = n! / (x! (n - x)!); calculates the number of ways to choose x successes from n trials.

Binomial Distribution

Probability distribution for a fixed number of independent trials, each with two possible outcomes (success or failure).

Poisson Distribution

Models count data in a fixed interval; used for calculating probabilities of a given number of events happening in a set timeframe.

Hypergeometric Distribution

For dependent trials without replacement, calculating probabilities of successes in a sample from a finite population.

Normal Distribution

A continuous probability distribution characterized by a symmetric bell shape, where the mean, median, and mode are equal.

Z-Score

Z = (X - μ) / σ; a measure of how many standard deviations an element is from the mean.

Empirical Rule

States that for a normal distribution, approximately 68% of data falls within one standard deviation from the mean.