1. MATH 256 Probability and Random Processes Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr 14/10/2011Lecture 3 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE 04 Random Variables Fall 2011

2. Probability Mass Function Is defined for a discrete variable X. 4/10/2011Lecture 32 Suppose that Then since x must be one of the values x i,

3. Example of probability mass function 4/10/2011Lecture 33

4. Expectation of a random variable If X is a discrete random variable having a probability mass function p(x) then the expectation or the expected value of X denoted by E[X] is defined by 4/10/2011Lecture 34 In other words, Take every possible value for X Multiply it by the probability of getting that value Add the result.

5. Examples of expectation For example, suppose you have a fair coin. You flip the coin, and define a random variable X such that – If the coin lands heads, X = 1 – If the coin lands tails, X = 2 Then the probability mass function of X is given by 4/10/2011Lecture 35 Or we can write

6. Expectation of a function of a random variable To find E[g(x)], that is, the expectation of g(X) Two step process: – find the pmf of g(x) – find E[g(x)] 4/10/2011Lecture 36

7. 4/10/2011Lecture 37 Let X denote a random variable that takes on any of the values –1, 0, and 1 with respective probabilities Compute Solution Let Y = X 2. Then the probability mass function of Y is given by

8. Proposition 4.1 4/10/2011Lecture 38 If X is a discrete random variable that takes on one of the values x i, i ≥ 1 with respective probabilities p(x i ), then any real valued function g. Check if this holds for the previous example:

9. Proof of Proposition 4.1 4/10/2011Lecture 39

10. 4/10/2011Lecture 310

11. 4/10/2011Lecture 311

12. 4/10/2011Lecture 312

13. 4/10/2011Lecture 313

14. Variance 4/10/2011Lecture 314

15. Variance 4/10/2011Lecture 315

16. 4/10/2011Lecture 316

17. Some more comments about variance 4/10/2011Lecture 317

18. Bernoulli Random Variables 4/10/2011Lecture 318

19. Binomial Random Variables 4/10/2011Lecture 319

20. Poisson Random Variable 4/10/2011Lecture 320 Some examples of random variables that generally obey the Poisson probability law [that is, they obey Equation (7.1)] are as follows: 1. The number of misprints on a page (or a group of pages) of a book 2. The number of people in a community who survive to age 100 3. The number of wrong telephone numbers that are dialed in a day 4. The number of packages of dog biscuits sold in a particular store each day 5. The number of customers entering a post office on a given day 6. The number of vacancies occurring during a year in the federal judicial system 7. The number of α-particles discharged in a fixed period of time from some radioactive material

21. Expectation of the sum of r.v.s Start with this: It’s really quite easy to show that: And from this we show that 4/10/2011Lecture 321