1. ENEE 324: Conditional Expectation Richard J. La Fall 2004

2. Conditional Expectation Example: Toss a coin 3 times X = number of heads in 3 independent tosses Y = length of the longest run of heads Compute 3 1 2 3 3/8 1/8 1/4

3. Conditional Expectation Example #2: (Tom and Jenny): Compute Ans: In general is a deterministic number which can be computed from the given value of Similarly,

4. Conditional Expectation Example #2: (Tom and Jenny)  can be thought of as a function of the rv, i.e., given the value we can compute the value of the function  Similarly, Conditional expectation  A function of rv => a derived rv !!!!  = value of the function evaluated at  Since is a rv, we can calculate its PMF, expected value, etc.

5. Conditional Expectation Conditional expectation E [ X|Y ]  A function of rv Y (i.e, f ( Y ))  f(y) = E[X|Y=y]  PMF of rv E [ X|Y ] :  Expected value of rv E [ X|Y ]

6. Conditional Expectation In general, Example: Toss a coin 3 times X = number of heads in 3 independent tosses Y = length of the longest run of heads 3 1 2 3 3/8 1/8 1/4

7. Conditional Expectation 3 1 2 3 3/8 1/8 1/4

8. Independent RVs Recall that two events A and B are independent if Definition: Two discrete rvs X and Y are independent if and only if for all  i.e., events { X=x } and { Y=y } are independent for all  Since

9. Independent RVs Example: Roll two six-sided dice X = number of dots on die #1 Y = number of dots on die #2 x y X and Y are independent

10. Independent RVs Example #2: Toss a coin 3 times X = number of heads in 3 independent tosses Y = maximum number of consecutive heads 3 1 2 3 3/8 1/8 1/4 X and Y not independent

11. Useful Fact In general, X, Y independent => X, Y uncorrelated ( Cov(X,Y) = 0 ) However, the converse is not true in general !

12. Useful Fact Example: Uncorrelated but not independent rvs 1 2 0.2 21 X and Y NOT INDEPENDENT !!

13. Multiple Discrete RVs Suppose be N rvs defined on the same underlying experiment Definition: Joint PMF

14. Multiple Discrete RVs Example: Suppose that the instructor plays a tennis match with Anna Kournikova. Let be the number of games that the instructor wins in set 1, 2, 3, respectively. Definition: Marginal PMF  Two RV case:

15. Multiple Discrete RVs Definition: Discrete rvs are independent if and only if for all Example: Roll N dice, and let be the number on the i -th die. where

16. Multiple Discrete RVs Functions of multiple rvs: Let  PMF:  Expected value:

17. Summary: Multiple Discrete RVs 1. Joint PMF of X and Y : 2. Margin PMF : 3. Function of RVs X and Y : PMF - Expected Value – 4. Conditional probability

18. Summary: Multiple Discrete RVs 5.Independent RVs

19. Problems Problem #1: The PMF for rvs H and B is given in the following table. Find the marginal PMFs and h=-1 h= 0 h= 1 b=0b=2b=4 0 0.4 0.2 0.1 0 0.1 0.1 0.1 0 h b 0.1 0.4 0.2 0.2 0.5 0.3 0.6 0.2 0.4 0.1 0.4 0.2

20. Problems: Problem #2: A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N 1 the number of tests made till the first defective is identified and by N 2 the number of additional tests until the second defective is identified. Find the joint PMF of N 1 and N 2.

21. Problems: Rvs X and Y have the joint PMF as shown. Let B = {X + Y <= 3}. Find the conditional PMF of X and Y given B. 3 1 2 3 1/8 1/4 1/16 1/12 1/8 1/12 1/16 12 3 1 2 3 1/8 1/4 1/8 1/12 1/16 12 Normalize by P(B) = 35/48

22. Problems: The marginal PMF of rv A is The conditional PMF of rv B given A is given by (a) Find the joint PMF of rvs A and B. (b) If B = 0, what is the conditional PMF ? (c) If A = 2, what the conditional expected value ?

23. Problems: Rvs X and Y have joint PMF given by the following matrix Are X and Y independent? Are they uncorrelated? y = -1y = 0 y = 1 00.250 0.250.250.25 x = -1 x = 1