General, Binomial, Poisson Probability Distributions

Random Variable

A variable that has a single numerical value, determined by chance, for each outcome of a procedure. May be discrete or continuous.

Examples of random variables

rolls of a dye, heads or tails on a coin, heights of random people, and scores on a standardized test

Probability Distribution

A description that gives the probability for each value of the random variable.

Population

Probability distributions are representative of the _______ not the sample

Probability Distributions Requirements

1. There is a numerical random variable, X, and its number values are associated with the corresponding probabilities

2. The sum of all probabilities must be equal to 1

3. Every individual probability must be between 0 and 1

Binomial Distribution

Describes the probability of “successes” in a specific number of trials. Allows us to deal with circumstances in which outcomes belong to two categories.

Examples of events with binomial distributions

number of plants (out of n) that are pollinated and number of animals that were caught in a camera trap.

Requirements of procedures leading to a binomial probability distribution

1. The procedure has a fixed number of trials (ex. two coin flips)

2. The trials must be independent, meaning that the outcome of any individual trial doesn’t affect the probabilities in the other trials.

3. Each trial must have all outcomes classified into exactly two categories

4. The probability of a success remains the same in all trials

Binomial Distribution Probability Formula

P(x) = (n!/(n-x)!*x!) p^x *q^n-x

n= number of trials

x= number of “successes” among n trials

p= probability of “success” in any one trial

q= probability of “failure” in any one trial; q=1-p

R Function: dbinom(x,n,p)

Mean formula for binomial distribution

mu = np

mu = population mean

n = # trials

p = probability

Variance formula for binomial distribution

sigma² = npq

Standard deviation formula for binomial distribution

sigma = sqrt(npq)

Range rule of thumb for identifying significant values

Significantly low: mu-2sigma or lower

Significantly high: mu+2sigma or higher

Values not significant: mu

Poisson Probability Distribution

A discrete probability distribution that applies to occurrences of some event over a specified interval

Specified Interval

Can be time, distance, area, volume, or some similar unit

Conditions for a Poisson Distribution

1. The random variable x is the number of occurrences of an event in some interval

2. The occurrences must be random

3. The occurrences must be independent of each other

4. The occurrences must be uniformly distributed over the interval being used

Probability formula for a Poisson Distribution

P(x) = (mu^x * e^-mu)/x!

e = 2.71828

mu = mean number of occurrences of the event in the intervals

x = specified number of events

Function in R dpois(x, mu)

True or False: The Poisson distribution is a special case of the binomial distribution and assumes a very large number of trials

True

Mean, Variance, and SD for Poisson Distribution

mean = mu

variance, sigma² = mu

SD, sigman = sqrt (mu)