Discrete Probability Distribution in Statistics


Discrete Probability Distribution

If the random variable can have only discrete outcomes such 1,2,3,4 5 etc, we have to use a discrete probability distribution.

There are 4 types of Discrete Distribution 
  1. Binomial or Bernoulli distribution
  2. Negative Binomial distribution
  3. Geometric Distribution
  4. Poisson Distribution

1. Binomial or Bernoulli distribution 

The probability distribution of a binomial random variable is called a binomial distribution.

The number of successes "x" in "n" repeated trials of a binomial experiment is called binomial random variable
  • Toss a coin it has only two outcome i.e. Head or Tail
  • Gender of Babies delivered in a hospital 
Properties of Binomial Distribution 
  • The experiment consists of n repeated trials.
  • Each trial can result in just two possible outcomes – Success or Failure
  • The probability of success, denoted by P, is the same on every trial.
  • Independent trials i.e. the outcome on one trial does not affect the outcome on other trials 
Consider the following statistical experiment. 

You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: 
  • The experiment consists of repeated trials. We flip a coin 2 times.
  • Each trial can result in just two possible outcomes - heads or tails.
  • The probability of success is constant - 0.5 on every trial.
  • The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
Formula for Binomial Distribution 
Where, x = Outcomes n = Trials p = probability of success on each trials

2. Negative Binomial distribution  

This type of distribution concerns the number of trials that must occur in order to have a predetermined number of successes or in other words it is concerns with the number of trials 'X' that must occur until we have 'r' successes.

3. Geometric Distribution 

Geometric distribution is a special case of negative binomial distribution where number of successes(r) is equal to 1

The experiment consists of a sequence of trials with the following conditions:
  • The trials are independent.
  • Each trial can result in one of two possible outcomes, success and failure.
  • The probability of success is the same for all trials.

4. Poisson Distribution

A statistical distribution showing the frequency probability of specific events when the average probability of a single occurrence is known. The Poisson distribution is a discrete function.

Properties of Poisson Experiments : 
  • The experiment results classified as successes or failures.
  • Average number of successes that occurs in a specified region is known.
  • Probability that a success will occur is proportional to the size of the region.
  • Probability that a success will occur in an extremely small region is virtually zero.
  • Events have to be counted as a whole number

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