presentation

Discrete and continuous distribution

Mentor- Dr. Richa Vatsa By...

Abhishek RanjanAlok Kumar PandeyArvind Kumar AlokAshwini Kr. Sameer

Random variables A random variable, usually written X, is

a variable whose possible values are numerical outcomes of a random phenomenon.

For example, tossing coin,shuffling cards, rolling dice.

There are two types of random variable, i.e. Discrete Random Variable and Continuous Random Variable

Types of Random Variable Discrete

Random Variable A discrete random

variable is a variable which can only take a countable number of values.

For example, throwing a dice can only take head values as 1 to 6 which is discrete.

Continuous Random Variable

A continuous random variable is a random variable where the data can take infinitely many values.

For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken..

PROBABILITY DISTRIBUTION OF DISCRETE RANDOM VARIABLE

Suppose x be a Discrete r.v on a sample space S of atmost countably infinite number of values x1,x2....xn with each possible outcome xi we associate a no. P(xi) i.e. Probability of xi. P(xi) must satisfy the conditions:(i) P(xi) ≥ 0, for all value of i.(ii) ∑P(xi)=1, i= 1,2....n

Types of Discrete distribution function

Binomial distribution. Geometric Distribution. Hypergeometric distribution. Negative binomial distribution. Poisson distribution.

Binomial Distribution

if the random variable X follows the binomialdistribution with parameters n ∈ ℕ and p ∈ [0,1],

we writeX ~ B(n, p). The probability of getting exactly k

successesin n trials is given by the probability mass

function :

POISSON DISTRIBUTION

A discrete random variable X is said to have a Poisson

distribution with parameter λ > 0, if, for k = 0, 1, 2, ...,

the probability mass function of X is given by

Geometric Distribution

A random variable X is defined to have Geometric Distribution if the density of X is given by :

P(X) = p(1-p)x

Mean = (1-p)/q

Variance = (1-p)/p2

It follows “memory loss property” in discrete

Hyper geometric Distribution

A random variable X follows the hypergeometric distribution if its probability mass function (pmf) is given by :

where N is the population size,

K is the number of success states in the population, n is the number of draws,

Continuous Random Variable

A continuous random variable is a random variable where the data can take infinitely many values.

For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken..

Continuous Probability Distribution

`If X is a continuous random variable, then function fx(.) in Fx(x) = is called the probability density function of X.

Types of Continuous Distribution

Normal distribution Gamma distribution Chi-square distribution Beta distribution Exponential distribution Uniform(continuous) distribution

Normal DistributionA random variable X is said to follow Normal

distribution if f(x)=

Mean= µVariance= Ϭ2

Standard Normal Distribution

A Normal distribution is said to be standard Normal distribution iff µ= 0 and Ϭ2=1 and

denoted by X~ N(0,1).

Linear combination of normal is also normal.

Any normal distribution of large sample can be reduced to standard normal by Z=X-µ/Ϭ

Exponential Distribution A random variable is said to be exponential dist. Iff f(x)= mean=1/λVariance=1/ λ 2

Exponential dist. Obeys the loss of memory property.It is used in life testing problem.No of inter arrival of customer and departure follows

exponential.

Uniform distribution A continuous r.v X is said to be uniform dist. Iff f(x)=1/b-a It is also known as rectangular dist. It does not depend upon value of x or r.v

irrespective of upper and lower limit value. Mean=(a+b)/2 Variance =(b-a)2/12 It is used in equiprobable situation. Mx(t)=ebt-eat/(b-a)t

BETA DISTRIBUTION A continuous random variable X is said

to be beta iff f(X)=

Mean=Variance=αβ/(α+β+1)(α+β)2

Beta dist are of two types beta of first kind and of 2nd kind .

Gamma distribution A continuous r.v. X is said to be gamma

distribution iff

Mean=α Variance=α The mean and variance of gamma are equal as

in case of poisson in discrete. Gamma tends to standard normal as for large

value of α. Sum of independent gamma is a gamma variate

which is known as linear combinatorial property.

Some relationship among distributions

Bernoulli dist. Follows binomial as the number of trials get increased i.e. Sum of bernoulli is binomial.

Binomial distribution tends to poisson as np=m and p→0 where m is finite value.

Binomial is an approximation of normal dist.

Normal tends to std normal. Sum of square of normal variate is chi-

square. Ratio of normal is cauchy dist.

References

Fundamental of Mathematical Statistics by S.C.Gupta and V.K. Kapoor

Introduction to theory of Statistics by Mood and Graybill

Operations Research by P.K. Gupta and D.S.Hira

Class Notes of Prof (Dr.) Shreekant Singh Class Notes of Dr. Richa vatsa

Questions or Suggestions

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