Just as a frequency distribution associates variable values to corresponding frequencies, a probability distribution associates the outcomes of a random experiment to corresponding probabilities.

### Discrete Probability Distribution Definition

An association of the outcomes of a discrete probability distribution with corresponding probabilities if the following two conditions are satisfied.
1. The sum of all the probabilities in the sample space = 1; that is ∑P(X) = 1
2. The probability of each event must be between 0 and 1. that is 0 ≤ P(X) ≤ 1.
The probabilities are found either theoretically or experimentally by observation.
The expectation of a discrete distribution E(X) = Mean μ = ∑X.P(X)
The variance of the discrete distribution X, σ2 = [E(x)]2- μ2.
Let us look at some of the discrete distributions.

## Binomial Distribution

The outcomes of a Binomial experiment along with their corresponding probabilities is called a Binomial Distribution.

A Binomial experiment is a random experiment consisting of a fixed number n of repeated independent trials. The outcomes of each trial can be reduced to two which can be considered as success and failure. The probability of success is the same in all n trials.

This discrete distribution gets the name binomial as the probabilities of the distribution can be related to the terms of a binomial expansion.

Binomial Probability Formula:
The probability of exactly X successes in n trials of a binomial experiment is given by
P(X: n, p) = C (n,X) . px . qn-x   where p = the probability of success in a single trial and q = 1 - p.
The binomial distribution is completely described by the two parameters n and p.

Example:
A fair coin is tossed three times, The discrete random variable X is defined as the number of heads turned.
The Binomial distribution for the experiment is shown below.

 X 0 1 2 3 P(X) $\frac{1}{8}$ $\frac{3}{8}$ $\frac{3}{8}$ $\frac{1}{8}$

Mean and Variance of a Binomial Distribution
Mean μ = np.          Variance σ2 = npq.

## Poisson Distribution

For large values of n, the computation of binomial probability involves heavy calculations.

Poisson Distribution is a discrete distribution which approximates the binomial distribution when n is large and p is small.

Poisson distribution is also used to describe the density distributed over time, area volume etc.

Poisson distribution is described by a single parameter λ which is the mean number of occurrences per unit.
The formula used for computing Poisson probability is,
P(X : λ) = $\frac{e^{-\lambda }\lambda ^{X}}{X!}$ where λ = np
The variance of a Poisson distribution = mean of the distribution = λ

### Solved Example

Question: It is estimated that 3% of University students are left handed. If there are 500 students in a College, What is the probability that exactly 10 of them are left handed.
Solution:

The info given in the problem are n = 500  p = 0.03.   Hence λ = np = 500 x 0.03 = 15

P( 10) = $\frac{e^{-15}15^{10}}{10!}$ ≈ 0.0486

Poisson distribution can also be approximated for a binomial distribution when the expected value np < 5.

## Geometric Distribution

In a Geometric Distribution the discrete random variable X can be viewed as the number of failures before the first success.

Geometric Probability formula
P(X : p) = qX . p where q = 1 - p.

The Geometric probabilities of the random variable X, form the infinite geometric series.

= p + pq + pq2 + + pqx + --------- = $\frac{p}{1-q}$ = $\frac{p}{p}$ = 1

$\sum_{1}^{\infty }$ P(X =x) = 1

### Solved Example

Question: A six sided die is thrown.  Find the probability that the number 6 turns out only in the tenth throw.
Solution:

p = $\frac{1}{6}$   q = $\frac{5}{6}$  and X = 9

P(X = 9) = ($\frac{5}{6}$)9 . ($\frac{1}{6}$) ≈ 0.0323