Understanding the Binomial and Poisson Distributions

These flashcards cover key concepts of binomial and Poisson distributions as outlined in the notes.

Binomial Distribution

A probability distribution that summarizes the likelihood that a value will take one of two independent outcomes across a specified number of trials.

Trial

Each repetition of an experiment in the context of a binomial distribution, typically resulting in two possible outcomes.

Factorial (k!)

The product of all positive integers up to k, denoted as k! = k(k-1)\dots3 \cdot 2 \cdot 1, with 0! defined as 1.

Binomial Coefficient

Denoted as {n \choose x}, it represents the number of ways to choose x successes from n trials, calculated using the formula n!/(x!(n-x)!). Alternatively, this can be represented as {n \choose k} = \frac{n!}{k!(n-k)!}. Also known as the combination formula.

Bernoulli Trials

Trials that meet three conditions: two possible outcomes (success and failure), independent outcomes, and a constant probability of success (denoted p).

Binomial Probability Formula

The formula P(X = x) = {n \choose x} p^x (1-p)^{n-x} used to calculate the probability of obtaining exactly x successes in n Bernoulli trials.

Mean (\mu) of Binomial Distribution

Calculated as \mu = n \cdot p, representing the expected number of successes.

Standard Deviation (\sigma) of Binomial Distribution

Calculated as \sigma = \sqrt{np(1-p)}, providing a measure of the variability of the number of successes.

Poisson Distribution

A discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.

Poisson Probability Formula

Given by P(X = x) = (e^{-\lambda} \cdot \lambda^x) / x! for x = 0, 1, 2, \dots, modeling the probability of x events in an interval given an average rate \lambda.

Mean (\mu) of Poisson Distribution

Equal to the parameter \mu = \lambda representing the average number of occurrences in a fixed interval.

Standard Deviation (\sigma) of Poisson Distribution

Equal to the square root of the mean, $