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Lecture 14 Simulation and Modeling

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

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Lecture 14 Simulation and Modeling

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  1. Lecture 14Simulation and Modeling Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411

  2. 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

  3. Discrete Uniform Distribution Probability Mass Probability Distribution

  4. 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

  5. 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

  6. 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 !!!

  7. One Experiment 2 red, 3 blue balls …

  8. Another Experiment 4 red, 1 blue balls …

  9. 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 ….

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

  11. Binomial Distribution Distribution

  12. 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)

  13. 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

  14. 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…

  15. 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 ??

  16. 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

  17. Geometric Distribution • Mean • Variance • MLE :

  18. 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)

  19. 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

  20. Negative Binomial Distribution • Mean : • Variance:

  21. Negative Binomial Distribution

  22. 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

  23. Poisson Distribution • Mean : • Variance: • MLE :

  24. 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)

  25. Poisson Distribution • If X1, X2, … .Xm are independent Random variables and Xi ~ Poisson (λi), • Then • X1+ X2 + X3 …. Xm ~ Poisson (λ1 +λ2 … +λm)

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