Distributions in Discrete Systems
engineering fundamentals Distributions in Discrete Systems
Directory | Career | News | Standards | Industrial | SpecSearch®
Statistics
Set Theory
Permutations & Combinations
Probability Theory
Distributions
  Discrete Systems
    Binomial
      Binomial Plot
    Hypergeometric
      Hypergeometric Plot
    Poisson
      Poisson Plot
  Continuous Systems
    Normal
      Normal Plot
    Uniform
      Uniform Plot
    Exponential
      Exponential Plot
Reliability
Least Squares
Resources
Bibliography


Login

Home Membership Magazines Forum Search Member Calculators

Materials

Design

Processes

Units

Formulas

Math
Binomial Distribution

The binomial distribution, also known as Bernoulli distribution, describes the random sampling processes that all outcomes are either yes or no (success/failure) without ambiguity. In addition, the probability will not change from one trial to another (independent). In this case, the number of total available samples remains the same throughout the sampling process. In other words, the total number of available samples is either infinity or the chosen sample are always placed back to the sampling pool before the next trial (Sampling with replacement).

Suppose that the probability of success in a single trial is p in a random sampling process and the failure rate is q, where q=1-p. The binomial distribution with exactly x successes in n trials, where x<=n, has the following important properties. See plots of binomial distributions.

Binomial Distribution
Density Function f(x)
Distribution Function F(x)
Mean mu
Variance sigma^2
Standard Deviation sigma^2

Hypergeometric Distribution

The hypergeometric distribution describes the random sampling processes that once a sample has been chosen, it will NOT be placed back to the sampling pool (sampling without replacement). In this case, the number of total available samples will decrease by one after each trial.

Suppose there are s successes and r failures in a sampling pool with N available samples (N=s+r). The hypergeometric distribution with exactly x successes in n trials, where x<=n<=N, has the following important properties. See plots of hypergeometric distributions.

Hypergeometric Distribution
Density Function f(x)
Distribution Function F(x)
Mean mu
Variance sigma^2
Standard Deviation sigma^2

where p=s/N and q=r/N=1-p.

Poisson Distribution

The Poisson distribution describes the random sampling process that the desired outcomes occur relatively infrequently but at a regular rate.

Suppose there are on average lambda successes in a large number of trials (large sampling period). The Poisson distribution with exactly x successes in the same sampling period has the following important properties. See plots of Poisson distributions.

Poisson Distribution
Density Function f(x)
Distribution Function F(x)
Mean mu
Variance sigma^2
Standard Deviation sigma^2

where lambda=np.

Home  Membership  About Us  Privacy  Disclaimer  Contact  Advertise

Copyright © 2014 eFunda, Inc.