الإحصاء و الاحتمالات (عرض 263) Probability and Statistics

1. Course code MSC 263 Class Days Sunday, Wednesday Credit hours 3hrs Instructor A. Sultan Email amsultan_52@yahoo.com Site www.freewebs.com/amsultan_52 الإحصاء و الاحتمالات (عرض 263)Probability and Statistics Dr. Ahmed M. Sultan

2. Course Grading Policy Quiz1 wk4 5 pts MT wk9 15 pts Quiz2 wk11 5 pts Hw 15 pts Project 10 pts Final 50 pts الإحصاء و الاحتمالات (عرض 263)Probability and Statistics Dr. Ahmed M. Sultan

3. Topics • Review of probability theory • Random Variables • Conditional probability and conditional expectation • The analysis of variance • Introduction • Single factor ANOVA • Simple linear regression and correlation • Introduction • The simple linear regression model • Estimation model parameters • Inferences about the slope parameters Dr. Ahmed M. Sultan

4. Topics (Cont.) • Multivariate regression analysis • When to use multivariate regression • Control variables • Interpreting coefficients • Goodness of fit (R squared statistic) • The exponential distribution and the Poisson process • Queueing theory • The M/M/1 queue • Steady state probabilities • Some performance measures Dr. Ahmed M. Sultan

5. Topics (Cont.) • The M/M/m queue • Steady state probabilities • Some performance measures • The M/M/1/K queue • Steady state probabilities • Some performance measures • Discrete event simulation • Generating pseudo random numbers • Congruential methods for generating pseudo random numbers • Composite generators • Statistical tests for goodness of fit Dr. Ahmed M. Sultan

6. Topics (Cont.) • Generating stochastic variables • The inverse transformation method • Sampling from continuous probability distribution • Data manipulation in MINITAB • Recording and transforming variables • Graphs and charts • Scatter plots • Histograms • Box plots and other charts • Cross tabulation Dr. Ahmed M. Sultan

7. References • Devore, J. “Probability and statistics for engineering and sciences” • Andrews Willing “A short introduction to queueing theory” • Banks, et. al. “Discrete event simulation” Dr. Ahmed M. Sultan

8. Review of probability theory • Laws of probability DEFINITION If an event E occurs m times in an n trial experiment, then the probability P(E) is defined as: i.e the experimrnt is repeated infinitely Dr. Ahmed M. Sultan

9. e.g In case of flipping a coin, the longer the experiment is repeated the closer will be the estimate to P(H) (or P(T)) to the theortical value of 0.5 0≤P(E) ≤1 P(E)=0 … E is impossible P(E)=1 … E is certain (sure) Dr. Ahmed M. Sultan

10. HW • In a study to correlate senior year high school students scores in mathematics and enrollment in engineering colleges a 1000 students were surveyed: 400 have studied mathematics Engineering enrollment shows that of the 1000 seniors: 150 have studied mathematics 29 have not Determine the probability of: • A student who studied mathematics is enrolled in engineering • A student who neither studies mathematics nor enrolled in engineering • A student is not studying engineering Dr. Ahmed M. Sultan

11. 1.1 Addition law of probability EUF … Union of E and F EF … Intersection of E and F If EF = ɸ, E and F are mutually exclusives or disjointed (occurrence of one precludes the other) • Addition law Dr. Ahmed M. Sultan

12. Example Rolling a die S={1,2,3,4,5,6} sample space P(1)= P(2)= P(3)= P(4)= P(5)= P(6)=1/6 Define E={1,2,3, or 4} F={3, 4, or 5} EF={3,4} P(E)= P(1)+ P(2)+ P(3)+ P(4)=4/6=2/3 P(F)=3/6=1/2 P(EF)=2/6=1/3 P(EUF)=P(E)+P(F)-P(EF) =2/3+1/2-1/3=5/6 Which is intuitively clear since EUF={1,2,3,4,5} Dr. Ahmed M. Sultan

13. HW • A fair die is tossed twice. E and F represent the outcomes of the two tosses. Compute the following probabilities • Sum of E and F is 11 • Sum of E and F is even • Sum of E and F is odd and greater than 3 • E is even less than 6 and F is odd greater than 1 • E is graeter than 2 and F is less than 4 • E is 4 and sum of E and F is odd Dr. Ahmed M. Sultan

14. 1.2 Conditional Probability • The conditional probability of an event E is the probability that the event will occur given the knowledge that an event F has already occurred. This probability is written P(E|F), notation for the probability of E given F. • In the case where events E and F are independent (where event F has no effect on the probability of event E), the conditional probability of event E given event F is simply the probability of event E, that is P(E). Dr. Ahmed M. Sultan

15. …(Cont.) • If events E and F are not independent, then the probability of the intersection of E and F (the probability that both events occur) is defined by P(E F) = P(F)P(E|F). • From this definition, the conditional probability P(E|F) is easily obtained by dividing by P(F): • P(E|F) = P(EF) / P(F) , P(F) > 0 • Note: This expression is only valid when P(F) is greater than 0. Dr. Ahmed M. Sultan

16. Example In rolling a die, what is the probability that the outcome is 6, given that the rolling turned up an even number Solution E={6}, F={2,4,6} thus P(E|F)=P(EF)/P(F)=P(E)/P(F)=(1/6)/(1/2)=1/3 Note that P(EF)=P(E) because E is a subset of F Dr. Ahmed M. Sultan

17. Example Ninety percent of flights depart on time. Eighty percent of flights arrive on time. Seventy-five percent of flights depart on time and arrive on time. (a) You are meeting a flight that departed on time. What is the probability that it will arrive on time? (b) You have met a flight, and it arrived on time. What is the probability that it departed on time? (c) Are the events, departing on time and arriving on time, independent? Dr. Ahmed M. Sultan

18. Solution Denote the events, A = { arriving on time} , D = {departing on time} . P{A} = 0.8, P{D} = 0.9, P{AD} = 0.75. (a) P{A I D} =P{AD} / P{D} = 0.75 / 0.9 = 0.8333 (b) P{D I A}= P{AD} / P{A} = 0.75 / 0.8 = 0.9375 (c) Events are not independent because P{AI D} ≠ P{A}, P{DI A} ≠ P{D}, P{AD} ≠ P{A}P{D}. Actually, anyone of these inequalities is sufficient to prove that A and D are dependent. Further, we see that P{AI D} > P{A} and P{D I A} > P {D}. In other words, departing on time increases the probability of arriving on time, and vise versa. This perfectly agrees with our intuition. Dr. Ahmed M. Sultan

19. HW In the example of rolling a die if given that the outcome is less than 6, determine probability of getting : • an even number • an odd number larger than 1. Dr. Ahmed M. Sultan

20. HW • You can toss a fair coin up to 7 times. You will win 1000 SR if three tails appear before a head is encountered. What are your chances of wining? Dr. Ahmed M. Sultan