# 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