1. Probability distribution Dr. Deshi Ye College of Computer Science, Zhejiang University yedeshi@zju.edu.cn

2. Outline  Chebyshev’s Theorem  The Poisson distribution & process  The Geometric Distribution  The Multinomial distribution  Simulation

3. 4.5 Chebyshev’s theorem  EX. The number of customers who visit a car dealer’s showroom on a Saturday morning is a random variable with mean 18 and standard variance 2.5.  Question: What probability can we assert that there will be more than 8 but fewer than 28 customers?

4. Chebyshev’s theorem  Theorem 4.1. If a probability distribution has mean and standard deviation, the probability of getting a value which deviates from by at least is at most Symbolically

5. Proof. Since

6. Corollary

7. EX  The number of customers who visit a car dealer’s showroom on a Saturday morning is a random variable with mean 18 and standard variance 2.5.  Question: What probability can we assert that there will be more than 8 but fewer than 28 customers?

8. Solution of EX  Let X be the number of customers. Hence k = 4

9. 4.6 Poisson Distribution  Poisson distribution: often serves as a model for counts which do not have a natural upper bound.  Poisson distribution

10. Comparing Poisson and binomial  It is known that 5% of the books bound at a certain bindery have defective bindings. Find the probability that 2 of 100 books bound by this bindery will have defective bindings using  A) the formula for the binomial distribution  B) the Poisson approximation to the binomial distribution

11. Solution  A) x=2, n=100, p=0.05 hence  B) x=2, for the Poisson distribution, we

12. Poisson approximation to binomial distribution EX: P129

13. Poisson distribution  Mean and variance of Poisson distribution Proof.

14. 4.7 Poisson Processes  On average 0.3 customer/min at a cafeteria,  Question: then what is the probability that 3 customers will arrive in 5- minute span?

15. Random process  Random process: is a physical process that is wholly or in part controlled by some sort of chance mechanism.  Characterize: time dependence, certain events do or do not take place at regular intervals of time or throughout continuous intervals of time.

16.  Goal: find the probability of x success during a time interval of length T.  Divide T into n equal parts of length ∆t T=n ∆t 1) The prob. of a success during a very small interval ∆t is given by a∆t 2) The prob. of more than one success during a small time interval ∆t is negligible. 3) The prob. of a success during such a time interval does not depend on what happen prior to that time.

17. By Poisson probability  When n is large enough the probability of x success during the time interval T is given the corresponding Poisson distribution with the parameter

18. Solution of EX  Solution: Consequently

19. 4.8 The Geometric Distribution  Suppose that a sequence of trials we are interested in the number of the trials on which the first success occurs.  Geometric distribution

20. Mean and variance  Mean of geometric distribution  Variance of geometric distribution

21. 4.9 The multinomial distribution Expansion of

22. 4.10 Simulation  Simulation can be useful tools for finding approximate probabilities for situation in which the actual probabilities are too difficult to calculate.  Approximation will typically be better the more repetitions you perform.

23. Activities  Divide students into groups such that each group has 3 students and also have 3 cards. 1) 3 cards written their own names. 2) Randomly shuffle 3) Each one pick cards, count number of matches TrialNameWhose nameNumber of matches 1Amy Barb Carol Barb Amy Carol 1

24. Theoretical analysis  XYZ, ABC, match X->A, Y->B, C->Z ABC ACB BAC BCA CAB CBA 3 1 1 0 0 1 Probability 0 1 2 3 1/3 ½ 0 1/6

25. 4-Person Analysis 01234 prob9/248/246/2401/24

26. Case study  The Coupon Collector’s problem