LECTURE 14 SIMULATION AND MODELING Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411

LECTURE 14

SIMULATION AND MODELINGMd. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET

CSE 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

1

1)(

ijxp

2

)( ji

12

1)1( 2 ij

Discrete Uniform Distribution

Probability Mass Probability Distribution

otherwise ,0

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

1)(

jiiijxP

jx

jxiij

ixix

xF

if , 1

if ,1

1 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 ppx

nxp

)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 :

p

p1

2

1

p

p

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

x

xsxp

p

ps )1(

2

)1(

p

ps

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)