Lecture 14 Simulation and Modeling - PowerPoint Presentation

Slide1

Lecture 14Simulation and Modeling

Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET

CSE 411Slide2

Discrete Uniform Distribution

Uniform distribution inside a interval.Say a random variable is equally likely to take value between i and j inclusiveWhat is the probability that X = x where

Mean

Variance Slide3

Discrete Uniform Distribution

Probability Mass

Probability DistributionSlide4

Binomial Distribution

Number of successes in n independent Bernoulli trials with probability p of success in each trialRelation between bernoulli and binomial :Suppose a two-tailed experiment

Pick a ball from the urn :

Ball is either blue or red

So two tailed testPr { Blue } = 6/10 = 0.6

Pr { Red } = 4/10 = 0.4This is a bernoulli trialSlide5

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 Slide6

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 !!!Slide7

One Experiment

2 red

,

3 blue

balls …Slide8

Another Experiment

4 red

,

1 blue

balls …Slide9

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 ….Slide10

Binomial Dist.

Mass function for various value of p

n = 15

n = 5

P = 0.9, 0.5, 0.2Slide11

Binomial Distribution

DistributionSlide12

Binomial Distribution

Mean Variance If Y1, Y

2

, … Y

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

nIf X1, X2… X

m are independent RV and Xi ~ bin(ni

,p) then X1 + X2

+ … + X

m

~ bin(t

1

+t

2

+…….t

m

, p)Slide13

Binomial Distribution

The bin(n,p) distribution is symmetric if and only if p=1/2X~ bin(n, p) if and only if X ~ bin (n, 1-p)The bin(1,p) and Bernoulli(p) distributions are sameSlide14

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…Slide15

Geometric Distribution

From previous example Say blue ball = failureSay red ball = success

Say we have infinite urns.

Step 1 C = 0

Step 2 Take a new urnStep 3 We pic one ballStep 4 If the ball is red, we are done … Print C

Else If the ball is blue C = C + 1, goto step 2Now, what is the probability that C will be 5 ?? Or 3 ?? Or 0 ??Slide16

Geometric Distribution

Probability of x failures= x blue balls followed by 1 red ballSo

x times failure

(1-p) to the power x

Followed by 1

successSlide17

Geometric Distribution

Mean Variance

MLE : Slide18

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 pThe 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)Slide19

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 inventorySlide20

Negative Binomial Distribution

Mean :

Variance: Slide21

Negative Binomial DistributionSlide22

Poisson Distribution

Number of events that occur in an interval of time when the events are occuring at a constant rateNumber of items in a batch of random sizeNumber of items demanded from an inventorySlide23

Poisson Distribution

Mean :

Variance

:

MLE : Slide24

Poisson Distribution

If Y1, Y2 …. be a sequence of non negative IID random variables and letThen the distribution of the Yi‘If and only if X ~ Poisson(λ

)Slide25

Poisson Distribution

If X1, X2, … .Xm are independent Random variables and Xi ~ Poisson (

λ

i

),Then X1

+ X2 + X3 …. Xm ~ Poisson (

λ1 +

λ2 … +λ

m

)