CPS Chapter 2: Discrete and Continuous Random Variables

Chapter 2 Concepts of Probability and Statistics

Bernoulli Distribution

Models single trial with two outcomes.

Binomial Distribution

Successes in fixed trials with constant probability.

Negative Binomial Distribution

Trials needed to achieve a fixed number of successes.

Geometric Distribution

Trials until first success; memoryless property.

Hypergeometric Distribution

Probability without replacement from finite populations.

Poisson Distribution

Models rare events over a fixed interval.

Degenerate Distribution

Single outcome with zero variance.

Discrete Uniform Distribution

Equal probability for all outcomes in a finite set.

Cumulative Distribution Function

Probability that a variable falls below a certain value.

Continuous Uniform Distribution

Equal likelihood across a continuous range.

Exponential Distribution

Time until first occurrence of an event.

Gamma Distribution

Waiting time for multiple events to occur.

Normal Distribution

Symmetrical bell curve defined by mean and standard deviation.

Beta Distribution

Defined on intervals with two shape parameters, the Beta distribution is used to model continuous random variables whose range is between 0 and 1. It is particularly useful in Bayesian statistics, modeling proportions, and in scenarios where outcomes are constrained within a bounded interval. The shape of the Beta distribution can vary widely based on its parameters, allowing for different forms such as uniform, U-shaped, or J-shaped distributions.

Weibull Distribution

Models various failure rates over time.

Mean

Average value of a distribution.

Variance

Measure of dispersion in a distribution.

Central Limit Theorem

As the size n of a simple random sample increases, the shape of the sampling distribution of x̄ tends toward being normally distributed.

Probability Density Function

Function representing probabilities of continuous outcomes.

Quantiles

Values dividing a probability distribution into intervals.

Memoryless Property

Future probabilities independent of past events.

Independent Trials

Outcomes of trials do not affect each other.

Fixed Probability

Probability remains constant across trials.