PROBABILITY AND STATISTICS FOR ENGINEERING

1. PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Sharif University of Technology Spring 2008

2. Text Book A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes, 4th Edition, McGraw Hill, 2000 Sharif University of Technology

3. TABLE OF CONTENTS PROBABILITY THEORY Lecture – 1 Basics Lecture – 2 Independence and Bernoulli Trials Lecture – 3 Random Variables Lecture – 4 Binomial Random Variable Applications, Conditional Probability Density Function and Stirling’s Formula. Lecture – 5 Function of a Random Variable Lecture – 6 Mean, Variance, Moments and Characteristic Functions Lecture – 7 Two Random Variables Lecture – 8 One Function of Two Random Variables Lecture – 9 Two Functions of Two Random Variables Lecture – 10 Joint Moments and Joint Characteristic Functions Lecture – 11 Conditional Density Functions and Conditional Expected Values Lecture – 12 Principles of Parameter Estimation Lecture – 13 The Weak Law and the Strong Law of Large numbers STOCHASTIC PROCESSES Lecture – 14 Stochastic Processes - Introduction Lecture – 15 Poisson Processes Lecture – 16 Mean square Estimation Lecture – 17 Long Term Trends and Hurst Phenomena Lecture – 18 Power Spectrum Lecture – 19 Series Representation of Stochastic processes Lecture – 20 Extinction Probability for Queues and Martingales Source AT: www.mhhe.com/papoulis

4. Teaching Assistants: Hoda Akbari Course Evaluation: Homeworks: 10-20% Midterm: 25-30% Final Exam: 50-60% Sharif University of Technology

5. Probability And Statistics Source: http://ocw.mit.edu Sharif University of Technology

6. PROBABILITY THEORY 1.Basics

7. Random Phenomena, Experiments • Study of random phenomena • Different outcomes • Outcomes that have certain underlying patterns about them • Experiment • repeatable conditions • Certain elementary events Ei occur in different but completely uncertain ways. • probability of the event Ei: P(Ei )>=0 Sharif University of Technology

8. Probability Definitions • Laplace’s Classical Definition • without actual experimentation • provided all these outcomes are equally likely. Example • a box with n white and m red balls • elementary outcomes: {white , red} • Probability of “selecting a white ball”: Sharif University of Technology

9. Probability Definitions • Relative Frequency Definition • The probability of an event A is defined as • nA is the number of occurrences of A • n is the total number of trials Sharif University of Technology

10. Probability Definitions Example • The probability that a given number is divisible by a prime p: Sharif University of Technology

11. Counting - Remark • General Product Rule if an operation consists of k steps each of which can be performed in ni ways (i = 1, 2, …, k), then the entire operation can be performed in ni ways. • Number of PINs • Number of elements in a Cartesian product • Number of PINs without repetition • Number of Input/Output tables for a circuit with n input signals • Number of iterations in nested loops Example Sharif University of Technology

12. Permutations and Combinations - Remark • If order matters choose k from n: • Permutations : • If order doesn't matters choose k from n: • Combinations : Example A fair coin is tossed 7 times. What is the probability of obtaining 3 heads? What is the probability of obtaining at most 3 heads? Sharif University of Technology

13. Example: The Birthday Problem • Suppose you have a class of 23 students. Would you think it likely or unlikely that at least two students will have the same birthday? • It turns out that the probability of at least two of 23 people having the same birthday is about 0.5 (50%). Sharif University of Technology

14. Axioms of Probability- Basics • The axiomatic approach to probability, due to Kolmogorov, developed through a set of axioms • The totality of all events known a priori, constitutes a set Ω, the set of all experimental outcomes. Sharif University of Technology

15. Axioms of Probability- Basics • A andBare subsets ofΩ. A B A A B Sharif University of Technology

16. Mutually Exclusiveness and Partitions • A and B are said to be mutually exclusive if • A partition of  is a collection of mutually exclusive(ME) subsets of  such that their union is . B A Sharif University of Technology

17. De-Morgan’s Laws A B A B A B B A Sharif University of Technology

18. Events • Often it is meaningful to talk about at least some of the subsets of  as events • we must have mechanism to compute their probabilities. Example • Tossing two coins simultaneously: • A: The event of “Head has occurred at least once”. Sharif University of Technology

19. Events and Set Operators • “Does an outcome belong to A or B” • “Does an outcome belong to A and B” • “Does an outcome fall outside A”? • These sets also qualify as events. • We shall formalize this using the notion of a Field. Sharif University of Technology

20. Fields • A collection of subsets of a nonempty set  forms a field Fif • Using (i) - (iii), it is easy to show that the following also belong to F. Sharif University of Technology

21. Fields If then • We shall reserve the term event only to members of F. • Assuming that the probabilityP(Ei )of elementary outcomesEiofΩare apriori defined. • The three axioms of probability defined below can be used to assign probabilities to more ‘complicated’ events. Sharif University of Technology

22. Axioms of Probability • For any event A, we assign a number P(A), called the probability of the event A. • Conclusions: Sharif University of Technology

23. Probability of Union of to Non-ME Sets A Sharif University of Technology

24. Union of Events • Is Union of denumerably infinite collection of pairwise disjoint events Aian event? • If so, what isP(A)? • We cannot use third probability axiom to compute P(A), since it only deals with two (or a finite number) of M.E. events. Sharif University of Technology

25. An Example for Intuitive Understanding • in an experiment, where the same coin is tossed indefinitely define: A = “head eventually appears”. • Our intuitive experience surely tells us that A is an event. If We have: • Extension of previous notions must be done based on our intuition as new axioms. Sharif University of Technology

26. σ-Field (Definition): • A field F is a σ-fieldif in addition to the three mentioned conditions, we have the following: • For every sequence of pairwise disjoint events belonging to F, their union also belongs to F Sharif University of Technology

27. Extending the Axioms of Probability • If Ai s are pairwise mutually exclusive • from experience we know that if we keep tossing a coin, eventually, a head must show up: • But: • Using the fourth probability axiom we have: Sharif University of Technology

28. Reasonablity • In previously mentioned coin tossing experiment: • So the fourth axiom seems reasonable. Sharif University of Technology

29. Summary: Probability Models • The triplet (, F, P) • is a nonempty set of elementary events • Fis a -field of subsets of . • P is a probability measure on the sets in F subject the four axioms • The probability of more complicated events must follow this framework by deduction. Sharif University of Technology

30. Conditional Probability • In N independent trials, suppose NA, NB, NAB denote the number of times events A, B and AB occur respectively. • According to the frequency interpretation of probability, for large N, • Among the NA occurrences of A, only NAB of them are also found among the NB occurrences of B. • Thus the following is a measure of “the event A given that B has already occurred”: Sharif University of Technology

31. Satisfying Probability Axioms • We represent this measure byP(A|B)and define: • As we will show, the above definition is a valid one as it satisfies all probability axioms discussed earlier. Sharif University of Technology

32. Satisfying Probability Axioms Sharif University of Technology

33. Properties of Conditional Probability Example • In a dice tossing experiment, • A: outcome is even • B: outcome is 2. • The statement that B has occurred • makes the odds for A greater Sharif University of Technology

34. Law of Total Probability • We can use the conditional probability to express the probability of a complicated event in terms of “simpler” related events. • Suppose that • So, Sharif University of Technology

35. Conditional Probability and Independence • A and B are said to be independent events, if • This definition is a probabilistic statement, not a set theoretic notion such as mutually exclusiveness. • If Aand Bare independent, • Thus knowing that the event B has occurred does not shed any more light into the event A. Sharif University of Technology

36. Independence - Example Example • From a box containing 6 white and 4 black balls, we remove two balls at random without replacement. • What is the probability that the first one is white and the second one is black? Sharif University of Technology

37. Example - continued • AreW1andB2independent? Removing the first ball has two possible outcomes: These outcomes form a partition because: So, Thus the two events are not independent. Sharif University of Technology

38. General Definition of Independence • Independence between 2 or more events: • Events A1,A2, ..., An are mutually independent if, for all possible subcollections of k ≤ n events: Example • In experiment of rolling a die, • A = {2, 4, 6} • B = {1, 2, 3, 4} • C = {1, 2, 4}. • Are events A and B independent? • What about A and C? Source: http://ocw.mit.edu Sharif University of Technology

39. Bayes’ Theorem We have: Thus, Also, Sharif University of Technology

40. Bayesian Updating: Application Of Bayes’ Theorem • Suppose thatA and Bare dependent events and A has apriori probability ofP(A). • How does Knowing that Bhas occurred affect the probability ofA? • The new probability can be computed based on Bayes’ Theorm. • Bayes’ Theormshows how to incorporate the knowledege about B’s occuring to calculate the new probability of A. Sharif University of Technology

41. Bayesian Updating - Example Example • Suppose there is a new music device in the market that plays a new digital format called MP∞. Since it’s new, it’s not 100% reliable. • You know that • 20% of the new devices don’t work at all, • 30% last only for 1 year, • and the rest last for 5 years. • If you buy one and it works fine, what is the probability that it will last for 5 years? Source: http://ocw.mit.edu Sharif University of Technology

42. Generalization of Bayes’ Theorem A more general version of Bayes’ theorem involves partition of Ω: In which, Represents a collection of mutually exclusive events with assiciated apriori probabilities: With the new information “B has occurred”, the information about Ai can be updated by the n conditional probabilities: Sharif University of Technology

43. Bayes’ Theorem - Example Example • Two boxes,B1andB2contain 100 and 200 light bulbs respectively. The first box has 15 defective bulbs and the second 5. Suppose a box is selected at random and one bulb is picked out. • What is the probability that it is defective? Sharif University of Technology

44. Example - Continued Suppose we test the bulb and it is found to be defective. What is the probability that it came from box 1? • Note that initially, • But because of greater ratio of defective bulbs in B1,this probability is increased after the bulb determined to be defective.. Sharif University of Technology