2. Random Variables and Stochastic Processes – 0903720 Dr. Ghazi Al Sukkar Email: ghazi.alsukkar@ju.edu.joghazi.alsukkar@ju.edu.jo Office Hours: will be posted soon Course Website: http://www2.ju.edu.jo/sites/academic/ghazi.alsukkar 1 Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul http://faraday.ee.emu.edu.tr/ee571 /

3. Cumulative Distribution Function (CDF) 2

4. 3

5. Properties of CDF 4

6. 5

7. Discrete-Type RV 6

8. Continuous-Type RV 7

9. Mixed RV. 8

10. Percentiles 9

11. Example The distribution function of a random variable X is given by Compute the following quantities ： (a) P(X < 2) (b) P(X = 2) (c) P(1  X < 3) (d) P(X > 3/2) (e) P(X = 5/2) (f) P(2<X  7) 10

12. Example For the experiment of flipping a fair coin twice, let X be the number of tails and calculate F(t), the distribution function of X, and then sketch its graph. Sol ： 11

13. Example Suppose that a bus arrives at a station every day between 10:00 Am and 10:30 AM, at random. Let X be the arrival time; find the distribution function of X, F(t), and then sketch its graph. Sol ： 12

14. Example The sales of a convenience store on a randomly selected day are X thousand dollars, where X is a random variable with a distribution function of the following form ： Suppose that this convenience store’s total sales on any given day are less than $2000. (a)Find the value of k. (b)Let A and B be the events that tomorrow the store’s total sales are between 500 and 1500 dollars, and over 1000 dollars, respectively. Find P(A) and P(B). (c)Are A and B independent events? 13

15. Probability Density Function (pdf) 14

16. 15

17. 16

18. Pdf for Discrete Random Variables 17

19. Example In the experiment of rolling a balanced die twice, let X be the maximum of the two numbers obtained. Determine and sketch the probability mass function and the distribution function of X. Sol ： 18

20. Example Can a function of the form be a probability mass function ? Sol ： 19

21. Example Let X be the number of births in a hospital until the first girl born. Determine the probability mass function and the distribution function of X. Assume that the probability is 1/2 that a baby born is a girl. Sol ： 20