1. Probabilistic and Statistical Techniques 1 Lecture 14 Dr. Nader Okasha

2. 2 Random Numbers

3. 3 Uncertainty, random events, and random variables:  Deterministic variables can assume a definite value.  A random variable may be defined with a range of possible values.  If X is a random variable, then: X = x is an event. X > x is an event. X < x is an event. x is a given value (outcome) of the random variable X

4. 4 L=10 R Deterministic variables R = 11 R = 11 > L = 10 No failure  The resistance has a definite value: Uncertainty, random events, and random variables: 11 is a given value (outcome) of the random variable R

5. 5 L=10 R RFailure? 11No 12No 9Yes 8No Random variables  The resistance may have any of these values: Uncertainty, random events, and random variables:

6. 6 L=10 R Random variables  The resistance may have any of these values: R Histogram Uncertainty, random events, and random variables:

7. 7 Random variables  The probabilities of the outcomes of the random variable can be determined from the frequencies of these outcomes Normal Distribution Probability Density Function R Uncertainty, random events, and random variables:

8. 8 Random variables  The frequencies can be used to establish a probability distribution that represents the relationship between the outcomes and their relative probabilities Normal Distribution Probability Density Function R=N(9.9962,0.9989) R Probability Density Function Uncertainty, random events, and random variables:

9. 9 Random variables  The probability distribution can then be used to determine the probabilities of the events Normal Distribution Probability Density Function R=N(9.9962,0.9989) Area =0.50115 R Probability Density Function Uncertainty, random events, and random variables:

10. 10 Random variables  Example: R<10 is the event of resistance higher than 10 (where the link fails) The probability of this event can be found by integration P(R<10) = 0.50115. Normal Distribution Probability Density Function R=N(9.9962,0.9989) Area =0.50115 R Probability Density Function Uncertainty, random events, and random variables:

11. 11 Random variables  Definition: A random variable is a mathematical vehicle for representing an event in analytical form Uncertainty, random events, and random variables:

12. 12

13. 13  Discrete  Continuous Uncertainty, random events, and random variables: Types of random variables

14. 14  Discrete random variable Either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process Example: The number of girls among a group of 10 people Uncertainty, random events, and random variables:

15. 15  Continuous random variable Infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions Example: The amount of water that a person can drink a day ; e.g. 2.343115 gallons per day Uncertainty, random events, and random variables:

16. 16 Example Identify the given random variables as being discrete or continuous: The no. of textbooks in a randomly selected bookstore The weight of a randomly selected a textbook The time it takes an author to write a textbook The no. of pages in a randomly selected textbook Uncertainty, random events, and random variables:

17. 17 Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) Discrete random variable with a finite number of values Uncertainty, random events, and random variables:

18. 18 Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... n Discrete random variable with an infinite sequence of values We can count the customers arriving, but there is no finite upper limit on the number that might arrive. Uncertainty, random events, and random variables:

19. 19 Examples of Discrete Random Variables Uncertainty, random events, and random variables:

20. 20 Continuous Random Variable Examples ExperimentRandom Variable (x)Possible Values for x Bank tellerTime between customer arrivals x >= 0 Fill a drink container Number of millimeters 0 <= x <= 200 Construct a new building Percentage of project complete as of a date 0 <= x <= 100 Test a new chemical process Temperature when the desired reaction take place 150 <= x <= 212 Uncertainty, random events, and random variables:

21. 21 Example TV Viewer Surveys: When four different households are surveyed on Monday night, the random variable x is the no. of households with televisions turned to Night Football on a specific channel xP(x) 0 0.522 1 0.368 2 0.098 3 0.011 4 0.001 Uncertainty, random events, and random variables:

22. 22 Example Paternity Blood Test: To settle a paternity suit, two different people are given blood tests. If x is the no. having group A blood, then x can be 0, 1, 2 and the corresponding probabilities are 0.36, 0.48, 0.16 respectively Uncertainty, random events, and random variables: