LECTURE 14SIMULATION AND MODELINGMd. Tanvir Al Amin, Lecturer, Dept. of CSE, BUETCSE

411

Discrete Uniform Distribution Uniform distribution inside a interval.

Say a random variable is equally likely to take value between i and j inclusive

What is the probability that X = x where

Mean

Variance

jxi

11)(

ij

xp

2)( ji

121)1( 2 ij

Discrete Uniform Distribution

Probability Mass Probability Distribution

otherwise ,0

},.....1,{ x if ,1

1)( jii

ijxP

jx

jxiijix

ix

xF

if , 1

if ,11

if , 0

)(

Binomial Distribution Number of successes in n independent

Bernoulli trials with probability p of success in each trial

Relation between bernoulli and binomial : Suppose a two-tailed experiment

Pick a ball from the urn : Ball is either blue or red So two tailed test Pr { Blue } = 6/10 = 0.6 Pr { Red } = 4/10 = 0.4 This is a bernoulli trial

Binomial Distribution Now suppose we have n such urns…

Pick a ball from urn 1, Pick a ball from urn 2, Pick a ball from urn 3…

All are independent events. Each of these parallel experiments have Pr{red}=0.4, and Pr{blue} = 0.6

Urn 1 Urn 2 Urn 3 Urn 4 Urn 5

Binomial Distribution All are independent events. Each of

these parallel experiments have Pr{red}=0.4, and Pr{blue} = 0.6

Does it mean, all of these experiments will have same outcome ?

NO !!!

One Experiment

2 red, 3 blue balls …

Another Experiment

4 red, 1 blue balls …

Binomial Distribution What is the probability that outcome is 1

red ball ? i.e. (4 blue balls) What is the probability that outcome is 3

red balls ? (and hence 2 blue balls)

Answer : Binomial Distribution…probability of x success in n independent two tailed tests ….

xnx ppxn

xp

)1()(

Binomial Dist.

Mass function for various value of p

n = 15n = 5P = 0.9, 0.5, 0.2

Binomial Distribution

Distribution

Binomial Distribution Mean Variance If Y1, Y2, … Yn are independent bernoulli

RV and Y is bin(n,p) then Y = Y1 + Y2 + …. Yn

If X1, X2… Xm are independent RV and Xi ~ bin(ni,p) then X1 + X2 + … + Xm ~ bin(t1+t2+…….tm, p)

np

)1( pnp

Binomial Distribution The bin(n,p) distribution is symmetric if

and only if p=1/2 X~ bin(n, p) if and only if X ~ bin (n, 1-p) The bin(1,p) and Bernoulli(p)

distributions are same

Geometric Distribution Number of failures before first success in

a sequence of independent Bernoulli trials with probability p of success on each trial…

The probability distribution of the number X of Bernoulli trials needed to get one success…

Geometric Distribution From previous example

Say blue ball = failure Say red ball = success Say we have infinite urns.

Step 1 C = 0 Step 2 Take a new urn Step 3 We pic one ball Step 4 If the ball is red, we are done … Print C

Else If the ball is blue C = C + 1, goto step 2 Now, what is the probability that C will be 5

?? Or 3 ?? Or 0 ??

Geometric Distribution Probability of x failures

= x blue balls followed by 1 red ball So

otherwise ,0

0 if ,)1()(

xppxp

x

x times failure(1-p) to the power xFollowed by 1

success

Geometric Distribution Mean

Variance

MLE :

pp1

2

1pp

1)(1

nXp

Geometric Distribution If X1, X2 … Xs are independent geom(p)

random variables, then X1 + X2 + … + Xs has a negative binomial distribution with parameters s and p

The geometric distribution is the discrete analog of the exponential distribution, in the sense that it is the only discrete distribution with the memoryless property.

The geom(p) distribution is a special case of the negative binomial distribution (with s=1 and the same value for p)

Negative Binomial Distribution Number of failures before the s-th

success in a sequence of independent bernoulli trials with probability p of success on each trial.

Number of good items inspected before encountering the s-th defective item

Number of items in a batch of random size

Number of items demanded from an inventory

Negative Binomial Distribution

Mean :

Variance:

otherwise ,0

0 xif ,)1(1

)(xs pp

xxs

xp

pps )1(

2

)1(pps

Negative Binomial Distribution

Poisson Distribution Number of events that occur in an

interval of time when the events are occuring at a constant rate

Number of items in a batch of random size

Number of items demanded from an inventory

Poisson Distribution

Mean : Variance: MLE :

otherwise ,0

0 if ,!)( xx

exp

x

)(nX

Poisson Distribution If Y1, Y2 …. be a sequence of non

negative IID random variables and let Then the distribution of the Yi‘ If and only if X ~ Poisson(λ)

}1:max{1

i

j

YjiX

}1{exp o

Poisson Distribution If X1, X2, … .Xm are independent Random

variables and Xi ~ Poisson (λi), Then X1+ X2 + X3 …. Xm ~ Poisson (λ1 +λ2 … +λm)