Discrete Distribution

Presented by:

Piyush TyagiRohit DeshmukhSagar MalikSanakarshan JoshiSayantan Banerjee

DISTRIBUTIONProbability distribution 

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

Random Variable: A numeric outcome that results from an experiment

Types of Distribution:1.Continuous Probability Distribution

Spread over an interval.Does not attain a specific value.

2.Discrete Probability DistributionWhose variables can take on only discrete value

Discrete DistributionAssign probability to each random variable.

A discrete distribution with probability function defined over k=1, 2, ..., has distribution function

Properties: 0≤P(xi) ≤1

Expected Value: Variance V(X):

𝑃ሺ𝑋𝑖ሻ= 1𝑛𝑖=0

µ=σ 𝑋𝑖𝑃(𝑋𝑖)𝑛𝑖=0

V(X)=σ (𝑋𝑖 − µ)2𝑃(𝑋𝑖)𝑛𝑖=0

Discrete Distribution contd….

Probability Distribution Function:

Shows probability of each ‘x’ value.

Cumulative Distribution Function:

Shows cumulative sum of probabilities.

Bernoulli Distribution:

It can result in one of 2 outcomes: Success or Failure.

Probability(Success)=π

Probability(Failure)=1-π

A Bernoulli random variable is the simplest random variable. It models an experiment in which there are only two outcomes.

Mean and Variance: For a Bernoulli random variable with success probability π :

Mean=πVariance=π(1- π)

James Bernoulli (Jacob I) born in Basel, SwitzerlandDec. 27, 1654-Aug. 16, 1705.

Binomial distribution:

Extension of Bernoulli’s experiment.

Arises when Bernoulli’s experiment is repeated n times.

Conditions for Binomial:

1. All trials should be independent.

2. All other conditions should remain same.

3. There are only two outcomes possible.

4. ‘π’ should not be too large or too small.

Binomial Distribution contd….

Properties: πx(1-π)n-x

PDF:

Mean: nπ

Standard Deviation:

𝑃ሺ𝑥ሻ= 𝑛!𝑥!ሺ𝑛−𝑥ሻ!πx(1-π)n-x

ඥ𝑛𝜋(1− 𝜋)

Poisson Distribution

Siméon Denis PoissonJune 21, 1781-April 25, 1840

It describes the number of occurrences within a randomly chosen unit of time. Necessary Condition-

Event must occur randomly and independently over a continuum period of time or space.

PDF-

P(x) =

Where,λ = mean arrivals per unit of time or spaceX= 0,1,2….

𝜆𝑥𝑒−𝜆𝑥!

Standard Deviation- ξ𝜆

λ

Poisson Distribution

Example: Mercy Hospital

Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?

λ=6/per hour= 3/per half-hour.

Ans: 0.168

Hyper geometric Distribution

Similar to binomial except sampling is without replacement.

Probability of each out come changes with each trial.

Parameter:N – Number of items in population.n – Number of items in a sample.s – Number of successes in population.

Properties:

PDF:

Mean: nπ where π=s/N

Standard Deviation : ඥ𝑛𝜋ሺ1− 𝜋ሻ x ට𝑁−𝑛𝑁−1

P(x)=൫𝑠𝑥൯൫𝑁−𝑠𝑛−𝑥൯

൫𝑁𝑛൯

Hypergeometric Distribution

Example: NevereadyBob Neveready has removed two dead batteries from a flashlight

and inadvertently mingled them with the two good batteries he intended as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?

n = 2 = number of batteries selected(sample size) N = 4 = number of batteries in total(population size) s = 2 = number of good batteries in total(success in population)

x = 2 = number of good batteries selected.

Ans: 0.167

Thank you