1 / 31

PROBABILITY AND STATISTICS FOR ENGINEERING

Learn about Normal and Poisson approximations in probability theory and discover applications in engineering problems. Understand conditional probability, Bayes' Theorem, and how to update information using statistics.

jontae
Download Presentation

PROBABILITY AND STATISTICS FOR ENGINEERING

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

  5. 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

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

  7. 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.

  8. The Poisson Theorem • If • Then

  9. 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

  10. 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 ,

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

  12. 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

  13. 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:

  14. 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

  15. 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

  16. Conditional Probability Density Function

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

  18. 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

  19. 1 1 Example - continued • For so that • For and

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

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

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

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

  24. 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:

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

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

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

  28. 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

  29. 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?

  30. 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.

More Related