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

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

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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 )