# Introduction

A probability distribution may be discrete probability distribution or continuous probability distribution.# Discrete Probability Distribution

A probability distribution is said to be discrete if the values taken by the corresponding random variables are discrete.

Following are the discrete theoretical distributions:

- Binomial distribution
- Poisson distribution

# Continuous Probability Distribution

A distribution is said to be continuous if the random variable takes any value, fractional or integral, in a specified interval.

Following are the continuous probability distributions:

- Normal distribution
- Chi-square distribution
- Students distribution (
*t*) *F*-distribution

# Binomial Distribution

Binomial distribution is associated with the French mathematician James Bernoulli.

Binomial probability distribution is a discrete probability distribution that applies in situation where there are fixed number of repeated trials in an experiment under identical conditions for which only one of the two mutually outcomes success or failure can result in each trial.

# Conditions Under which Binomial Distribution is Applied

- The experiment conducted must have only two mutually exclusive outcomes.
- The experiment must be repeated a fixed (finite) number of times.
- Each trial has only two outcomes, success and failure.
- Probabilities of the outcomes must be same for every trial.
- The outcomes must be independent from one trial to another. This means the outcome of one trial should not affect the outcome of any other trial.

# Properties of Binomial Distribution

It is a discrete probability distribution.

The parameters are â€˜*p*â€™ and â€˜*n*â€™, where *p* must be greater than 0 but less than 1, *i.e.*, 0 < *p *< 1

- Mean is given by
*E*(*x*) =*np* - Variance is
*Var(x)*=*npq**where,**q*= 1 â€“*p* - Standard deviation is
- If
*x*and*y*are two independent random variables following Binomial distribution with parameters (*n*_{1},*p*) and (*n*_{2},*p*), then*x*+*y*also follows binomial distribution with a parameter*B*(*n*_{1 }+*n*_{2},*p*). - A discrete random variable
*x*is said to follow binomial distribution, if its probability mass function is given by*P*(*x*=*r*) =Ã—^{n}C_{r}*p*(1 âˆ’^{r}*p*)^{n}^{âˆ’r},*r*= 0, 1, 2,...,*n*.*n*= total number of trials*p*= probability of success in a single trial - Mode of binomial distribution is given by Î¼ 0 = (n+1)p
*m*0 = (*n*+ 1)*p*, [(*n*+ 1)*p*]-1 if (*n*+ 1)*p*, is an integer*m*0 = Integral part of (*n*+ 1)*p*, if (*n*+ 1)*p*is not an integer - Maximum variance is at
*p*=*q*= 0.5 (*V*max =*n*/4) - Minimum variance (
*V*min) is at extreme point i.e, when*p*= 0 or*p*= 1

**Note: **

*n*and*p*are called the parameters of a*binomial distribution.*- A binomial distribution with parameters
*n*and*p*is denoted by*B*(*n*,*p*). - Binomial distribution is also known as bi-parametric as it has two parameters
*n*and*p*.

# Applications of Binomial

- Number of defectives in a lot size â€˜
*n*â€™ - Number of absentees in a class of â€˜
*n*â€™ persons - Number of married men in a group of â€˜
*n*â€™ men

# Shape of a Binomial Distribution

- If p = 0.5, it is symmetrical.
- If
*p*< 0.5, it is skewed to the right. - If
*p*> 0.5, it is skewed to the left.

**Note:** Mean > Variance > Standard deviation

Example

Brokerage survey reports that 30% of individual investors have used a discount broker, that is, one which does not charge the full commission. In a random sample of 9 individuals, what is the probability if

a. Exactly two of the sampled individuals have used a discount broker

b. Not more than 3 have used a discount broker

c. At least 3 of them have used a discount broker

a. Exactly two of the sampled individuals have used a discount broker

b. Not more than 3 have used a discount broker

c. At least 3 of them have used a discount broker

Solution

The probability that individual investors have used a discount broker is
âˆ´
a. Probability that exactly 2 of the 9 individuals have used a discount broker is given by

*p*= 0.30.*q*= 1 âˆ’*p*= 0.70b. Probability that out of the 9 randomly selected individuals not more than 3 have used a discount broker is given by
Probability that out of the 9 randomly selected individuals at least 3 have used a discount broker is given by

Example

The indices of occupational diseases in an industry are such that the workers have 20% of chance of suffering from it. What is the probability that out of the 6 workers, 4 or more workers will come in contact of diseases?

Solution

The probability of a worker suffering from the diseases is

*p*= 20/100 = 1/5 âˆ´

*q*= 1 -

*p*= 1 â€“ (1/5) = 4/5

The probability of 4 or more, (e.g., 6) coming in contact of the diseases is given by

Hence, the probability that out of 6 workers or more will come in contact of the diseases is 0.01695.

Example

The probability that an evening college student will graduate is 0.4. Determine the probability that out of 5 students (a) none (b) one and (c) at least one will graduate.

Solution

Given

a.

b.

c.

*p*= 0.4 and*q*= 0.6a.

*P*(*x*= no graduate) =^{5}*C*_{0}(0.4) (0.6)^{5}= 1 Ã— 1 Ã— 0.0777 = 0.0777b.

*P*(*x*= 1) =^{5}*C*_{0}(0.4)^{1}(0.6)^{4}= 0.2592c.

*P*(*x*â‰¥ 1) = 1 â€“*P*(*x*= 0) = 1 â€“ 0.0777 = 0.9223

Example

If on average 8 ships out of 10 arrive safely at a port, find the mean and standard deviation of the number of ships arriving safely out of total 1600 ships.

Solution

Probability of safe arrival,

Mean of ships arriving safely =

Standard deviation Ïƒ =

Hence, the Mean and standard deviation of the ships returning safely is respectively 1280 and 16.

*p*= 8/10 = 0.8 and*q*= 1 â€“*p*= 1 â€“ 0.8 = 0.2.Mean of ships arriving safely =

*m*=*np*= 1600 x 0.8 = 1280Standard deviation Ïƒ =

Hence, the Mean and standard deviation of the ships returning safely is respectively 1280 and 16.