PROBABILITY AND STATISTICS FOR ENGINEERING

1. PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology

2. Let X represent a Binomial r.v ,Then from • for large n. In this context, two approximations are extremely useful.

3. The Normal Approximation (Demoivre-Laplace Theorem) • If hen for k in the neighborhood of np, we can approximate (for example, when with p held fixed) • And we have: • where

4. As we know, • If and are within with approximation: • where • We can express this formula in terms of the normalized integral • that has been tabulated extensively.

6. Example • A fair coin is tossed 5,000 times. • Find the probability that the number of heads is between 2,475 to 2,525. • We need • Since n is large we can use the normal approximation. • so that and • and • So the approximation is valid for and Solution

7. Example - continued • Here, • Using the table,

8. The Poisson Approximation • For large n, the Gaussian approximation of a binomial r.v is valid only if p is fixed, i.e., only if and • What if is small, or if it does not increase with n? • for example when , such that is a fixed number.

9. The Poisson Theorem • If • Then

10. The Poisson Approximation • Consider random arrivals such as telephone calls over a line. • n : total number of calls in the interval • as we have • Suppose • Δ: a small interval of duration • Let be the probability of k calls in the interval Δ • We have shown that

11. The Poisson Approximation • p: probability of a particular call occurring in Δ: • as • Normal approximation is invalid herebecause is not valid. • : probability of obtaining k calls (in any order) in an interval of duration Δ ,

12. The Poisson Approximation • Thus, the Poisson p.m.f

13. Example: Winning a Lottery • Suppose • two million lottery tickets are issued • with 100 winning tickets among them. • a) If a person purchases 100 tickets, what is the probability of winning? Solution The probability of buying a winning ticket

14. Winning a Lottery - continued • X: number of winning tickets • n: number of purchased tickets , • P: an approximate Poisson distribution with parameter • So, The Probability of winning is:

15. Winning a Lottery - continued • b) How many tickets should one buy to be 95% confident of having a winning ticket? • we need • But or • Thus one needs to buy about 60,000 tickets to be 95% confident of having a winning ticket! Solution

16. Example: Danger in Space Mission • A space craft has 100,000 components • The probability of any one component being defective is • The mission will be in danger if five or more components become defective. • Find the probability of such an event. • n is large and p is small • Poisson Approximation with parameter Solution

17. Conditional Probability Density Function

18. Conditional Probability Density Function • Further, • Since for

19. 1 1 1 1 (b) (a) Example • Toss a coin and X(T)=0, X(H)=1. • Suppose • Determine • has the following form. • We need for all x. • For so that • and Solution

20. 1 1 Example - continued • For so that • For and

21. Example • Given suppose Find • We will first determine • For so that • For so that Solution

22. (b) (a) Example - continued • Thus • and hence

23. Example • Let B represent the event with • For a given determine and Solution

24. Example - continued • For we have and hence • For we have and hence • For we have so that • Thus,

25. Conditional p.d.f & Bayes’ Theorem • First, we extend the conditional probability results to random variables: • We know that If is a partition of S and B is an arbitrary event, then: • By setting we obtain:

26. Conditional p.d.f & Bayes’ Theorem • Using: • We obtain: • For ,

27. Conditional p.d.f & Bayes’ Theorem • Let so that in the limit as • or • we also get • or (Total Probability Theorem)

28. Bayes’ Theorem (continuous version) • using total probability theorem in • We get the desired result

29. Example: Coin Tossing Problem Revisited • probability of obtaining a head in a toss. • For a given coin, a-priori p can possess any value in (0,1). • : A uniform in the absence of any additional information • After tossing the coin n times, k heads are observed. • How can we update this is new information? • Let A= “k heads in n specific tosses”. • Since these tosses result in a specific sequence, • and using Total Probability Theorem we get Solution

30. Example - continued • The a-posteriori p.d.f represents the updated information given the event A, • Using • This is a beta distribution. • We can use this a-posteriori p.d.f to make further predictions. • For example, in the light of the above experiment, what can we say about the probability of a head occurring in the next (n+1)th toss?

31. Example - continued • Let B= “head occurring in the (n+1)th toss, given that k heads have occurred in n previous tosses”. • Clearly • From Total Probability Theorem, • Using (1) in (2), we get: • Thus, if n =10, and k = 6, then • which is more realistic compared to p = 0.5.