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Summary of Discrete Statistical Distributions

 A distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.

(Ref: Statistical Models in Engineering, by Gerald Hahn and Samuel Shapiro, John Wiley and Sons, 1967)

Statistical Concepts
Discrete Distrbutions
Binomial distribution
  • It gives the probability of exactly x successes in n independent trials. Here, the probability of success (p) on a single trial is constant. 
  •  When n=1 i.e. the experiment has only one trial, the Binomial distribution becomes Bernoulli distribution. 
  • For example, when we have to calculate the probability of 7 or more heads in 10 tosses of a fair coin. Notice that the probability of head at each trial is constant in this experiment.
  • Frequently used in quality control, reliability, survey sampling, and other industrial problems. 

Note: It can sometimes be approximated by Poisson or by Normal distribution.

Poisson distribution
  • It gives the probability of exactly x independent occurrences during a fixed period of time.
  • For example, it can represent the distribution of the number of defects in a piece of material, customer arrivals, insurance claims, incoming telephones, Alpha particles emitted, etc. 
  • Events take place independently and at a constant rate. 
  • Used frequently in quality control, reliability, queuing theory, and so on.

Note: It can be used as an approximation to the binomial distribution.

Multinomial distribution
  • It gives the probability of exactly xi outcomes of the event i (for i equal to 1,2,…,k) in n independent trials when the probability p of the event in a single trial is a constant.
  • For example, let 4 companies are bidding for each of the three contracts with specified success probabilities. What is the probability that a single company will receive all the orders?
  •  It has frequent use in quality control and other industrial problems.

Note: It is a generalization of the binomial distribution for more than 2 outcomes.

Hyper-geometric distribution
  • It gives the probability of getting exactly x good units in a sample of n units from a population of N units when there are k bad units in the population. 
  • For example, given a lot with 21 good units and 4 defectives what is the probability that a sample of five will yield not more than 1 defective?
  • It is used in quality control and related applications.

Note: It may be approximated by a binomial distribution when n is small relative to N.

Geometric distribution
  • It gives the probability of occurrence of exactly x Bernoulli trials before the first success. 
  • For example, you ask people outside a polling station who they voted for until you find someone that voted for the independent candidate in a local election.
  • It is used in quality control, reliability, and other industrial situations.
Negative Binomial distribution
  • In a sequence of independent Bernoulli(p) trials, let the random variable X denote the trial at which the rth success occurs, where r is a fixed integer. Then, we say that X has a negative binomial(r, p) distribution.
  • In other words, this distribution assists in modeling the number of trials that must occur in order to have a predetermined number of successes.
  • For example, Toss a fair coin until you get 8 heads. In this case, the 8 is the required number of successes. 

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