ENEE 324: Conditional Expectation

ENEE 324:Conditional Expectation Richard J. La Fall 2004

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 1/8 3 1/4 2 3/8 1/8 1 1/8 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,

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 => aderived rv !!!! • = value of the function evaluated at • Since is a rv, we can calculate its PMF, expected value, etc.

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]

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 1/8 3 1/4 2 3/8 1/8 1 1/8 3

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

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

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

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

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

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

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

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:

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

Multiple Discrete RVs • Functions of multiple rvs: Let • PMF: • Expected value:

Summary: Multiple Discrete RVs • Joint PMF of X and Y : • Margin PMF : • Function of RVs X and Y : • PMF - • Expected Value – • Conditional probability

Summary: Multiple Discrete RVs 5. Independent RVs

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

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 N1 the number of tests made till the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint PMF of N1 and N2.

3 Normalize by P(B) = 35/48 2 1/8 1/12 1 1/8 1/12 1/16 1/4 1 2 3 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. 1/16 3 1/12 1/16 2 1/8 1/12 1/16 1 1/8 1/12 1/16 1/4 1 2 3

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 ?

Problems: • Rvs X and Y have joint PMF given by the following matrix Are X and Y independent? Are they uncorrelated? y = -1 y = 0 y = 1 0 0.25 0 0.25 0.25 0.25 x = -1 x = 1

ENEE 324: Conditional Expectation