1. ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 6 – More Discrete Random Variables Farinaz Koushanfar ECE Dept., Rice University Sept 10, 2009

2. ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: Section 2.1-2.4, 2-6 Review Discrete random variables – Examples of PMFs: Binomial, Geometric – Expectation, mean, and variance – Conditioning

3. ELEC 303, Koushanfar, Fall’09 Review – random variable A random variable is defined by a deterministic function that maps from the sample space to real numbers

4. ELEC 303, Koushanfar, Fall’09 Mean

5. ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

6. ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

7. ELEC 303, Koushanfar, Fall’09 Mean (Expectation) Definition: Interpretations – Center of gravity for the PMF – Average in a large number of repetitions for one experiment Figure courtesy of http://www.elfwood.com

8. ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

9. ELEC 303, Koushanfar, Fall’09 Properties of expectation Let X be a RV and let Y=g(X) – It is often hard to calculate E[Y]=  y yP Y (y) – It is easier to compute: E[Y]=  x g(x)P X (x) Second moment: E[X 2 ] Generally speaking, E[g(X)]  g(E[X]) Variance:

10. ELEC 303, Koushanfar, Fall’09 Properties of Expectation If  and  are constants, and X and Y are RVs: – E[  ]= – E[  X]= – E[  X+  ]= – E[X+Y]= – E[X.Y]=

11. ELEC 303, Koushanfar, Fall’09 Variance

12. ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

13. ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

14. ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

15. ELEC 303, Koushanfar, Fall’09 Discrete uniform distribution

16. ELEC 303, Koushanfar, Fall’09 Average speed vs. average time If weather is good (probability=0.6) Alice walks the 2 miles with speed 5miles/hr. If weather is bad, Alice rides her motorcycle at a speed V=30 miles/hr. What is the mean of the time T to get to the class? Correct Solution - derive the PMF of T: P T (t)=0.6, if t=2/5; P T (t)=0.4, if t=2/30  E[T] = 0.6  2/5 + 0.4  2/30 = 4/15 hrs = 16 mins

17. ELEC 303, Koushanfar, Fall’09 Average speed vs. average time If weather is good (probability=0.6) Alice walks the 2 miles with speed 5miles/hr. If weather is bad, Alice rides her motorcycle at a speed V=30 miles/hr. What is the mean of the time T to get to the class? Mistake: it is wrong to find the ave speed E[V] = 0.6  5 + 0.4  30 = 15 miles/hr  E[T] = 2/E[V] = 2/15 hrs = 8 mins Summary: E[T] = E[2/V]  2/E[V]

18. ELEC 303, Koushanfar, Fall’09 Review: discrete random variable PMF, expectation, variance Probability mass function (PMF) P X (x) = P (X=x)  x P X (x)=1

19. ELEC 303, Koushanfar, Fall’09 Some properties of expectation

20. ELEC 303, Koushanfar, Fall’09 Bernoulli (indicator) RV

21. ELEC 303, Koushanfar, Fall’09 Binomial RV

22. ELEC 303, Koushanfar, Fall’09 Conditional PMF and expectation

23. ELEC 303, Koushanfar, Fall’09 Geometric PMF

24. ELEC 303, Koushanfar, Fall’09 Geometric PMF

25. ELEC 303, Koushanfar, Fall’09 Total expectation theorem

26. ELEC 303, Koushanfar, Fall’09 Geometric random variable

27. ELEC 303, Koushanfar, Fall’09 Geometric random variable