统计学重点整理CH5-Discrete Probability Distributions

CH5

Discrete vs. Continuous Distributions

Discrete distributions –

constructed from discrete (individually distinct) random variables Continuous distributions –

based on continuous random variables

Random Variable - a variable which contains the outcomes of a chance experiment

Categories of Random Variables

Discrete Random Variable - the set of all possible values is at most a finite or a countable infinite

number of possible values

Continuous Random Variable - takes on values at every point over a given interval

Discrete Distribution

Mean of discrete distribution – is the long run average

If the process is repeated long enough, the average of the outcomes will approach the long

run average (mean)

Requires the process to eventually have a number which is the product of many processes Mean of a discrete distribution

Variance and Standard Deviation of a Discrete Distribution

Binomial Distribution

Probability function

Mean value

Variance and Standard Deviation

Poisson Distribution

The Poisson distribution focuses only on the number of discrete occurrences over some interval

or continuum

Poisson does not have a given number of trials (n) as a binomial experiment does Occurrences are independent of other occurrences Occurrences occur over an interval

If Poisson distribution is studied over a long period of time, a long run average can be determined

The average is denoted by lambda (λ)

Each Poisson problem contains a lambda value from which the probabilities are determined A Poisson distribution can be described by λ

alone

where (X) is the long run average;

X = outcome, P = Probability of X

()∑-?=)(2

2

X P X μσ

()∑?==)(X P X

X E μ

Poisson Distribution

Probability function

Mean of a Poisson Distribution is λ

Variance of a Poisson distribution is also λ

Understanding the mean of a Poisson distribution gives a feel for the actual occurrences that are

likely to happen

Std Dev = Square root of λ

Poisson Approximation of the Binomial Distribution

Binomial probabilities are difficult to calculate when n is large

.

Under certain conditions binomial probabilities may be approximated by Poisson probabilities. Poisson approximation

Hypergeometric Distribution

Sampling without replacement

from a finite population The number of objects in the population is denoted N

.

Each trial has exactly two possible outcomes, success and failure.

Trials are not independent

X is the number of successes in the n

trials

The binomial is an acceptable approximation, if n < 5% N .

Otherwise it is not.

Probability function

N is population size

n is sample size

A is number of successes in population

x is number of successes in sample

.acceptable is ion approx imat the ,7 and 20 If ≤?>p n n . Use p n ?=λ