1. Discrete Random Variable

2. Outline Expected Value (Section 2.5) Functions of a Random Variable (Section 2.6) Expected Value of a Derived Random Variable (Section 2.7) Variance and Standard of Deviation (Section 2.8) Conditional Probability Mass Function (Section 2.9)

3. Expected Value

6. Expected Value of a Geometric Random Variable

7. Functions for generating random samples bernoullirv(p,m) binomialrv(p,m) geometricrf(p,m) pascalrv(k,p,m) poissonrv(alpha,m)

8. mean() in matlab mean() calculates the average of a vector. E[X] is not the same as mean(X) unless the length of the vector is sufficiently large.

9. Calculate the Expected Value of a Geometric RV in Matlab X is a geometrical random variable. Analytical calculation: E[X]=5 for p=0.2

10. Functions of a Random Variable

11. Derived Random Variable We observe sample values of a random variable and these sample values to compute other quantities Example: – We measure the power level of received signal in a cellular phone. An observation is x, the power level in units of milliwatts. – We convert the measurements to decibles by calculating y=10log 10 (x) dBm – X is a random variable – Y is a derived random variable because it is derived from X

12. 1 page: 10 cents 2 pages: 19 cents 3 pages: 27 cents 4 pages: 34 cents 5 pages: 40 cents Determine Y with a polynomial: Y=aX+bX 2 Determine a and b using X=1,3. Solve the equations in Mathematica or by hand.

17. Simulate P C (c) in matlab

18. Expected Value of a Derived Random Variable

19. Expected Value of a Derived Variable

20. Example Prove the following theorem:

22. Estimating Expected Monthly Phone Bill

23. Variance and Standard Deviation

24. Key Points An average is a typical value of a random variable. The next question: – “How typical is the average?” – “What are the chances of observing an event far from the average?” The variance of a random variable X describes the difference between X and its expected value.

25. Definitions

26. Properties of Variance/Standard of Deviation

28. Conditional Probability Mass Function

29. Definition/Theorems

32. If time permits…..

41. Matlab

42. Generating Random Variable

43. Derived Random Variable

44. Conditioning