Site Overlay # 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)

##### 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.

For any query, suggestion or feedback, please reach out to us on LinkedIn. You can find some of the resources that helped us here.

If you can contribute by talking about your interview experience, it will definitely create an impact. Please fill this form to help students get a perspective about the interview structure and questions.

You can read other articles here.

Cheers and Best!