**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

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

- Binomial or Bernoulli distribution
- Negative Binomial distribution
- Geometric Distribution
- 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

- 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

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

Author: Mohit T (Algae Study)

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