LECTURE 14 SIMULATION AND MODELING Md. 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

Discrete Uniform Distribution Probability MassProbability Distribution

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 1Urn 2Urn 3Urn 4Urn 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 ….

Binomial Dist. Mass function for various value of p n = 15n = 5 P = 0.9, 0.5, 0.2

Binomial Distribution Distribution

Binomial Distribution  Mean  Variance  If Y 1, Y 2, … Y n are independent bernoulli RV and Y is bin(n,p) then Y = Y 1 + Y 2 + …. Y n  If X 1, X 2 … X m are independent RV and X i ~ bin(n i,p) then X 1 + X 2 + … + X m ~ bin(t 1 +t 2 +…….t m, p)

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 3We 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 x times failure (1-p) to the power x Followed by 1 success

Geometric Distribution  Mean  Variance  MLE :

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:

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 :

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)

Poisson Distribution  If X 1, X 2, ….X m are independent Random variables and Xi ~ Poisson ( λ i ),  Then  X 1 + X 2 + X 3 …. X m ~ Poisson ( λ 1 +λ 2 … +λ m )