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

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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

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Discrete Uniform Distribution Probability MassProbability Distribution

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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

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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

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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 !!!

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One Experiment 2 red, 3 blue balls

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Another Experiment 4 red, 1 blue balls

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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 Distributionprobability of x success in n independent two tailed tests .

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

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Binomial Distribution Distribution

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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)

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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

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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

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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 ??

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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

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Geometric Distribution Mean Variance MLE :

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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)

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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

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Negative Binomial Distribution Mean : Variance:

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Negative Binomial Distribution

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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

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Poisson Distribution Mean : Variance: MLE :

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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( )

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