1. Chapter 7 Expectation 7.1 Mathematical expectation

2. 7.1 Mathematical Expectation Mathematical expectation =expected long run average Simulation 1: toss a fair coin H  1, T  0. n=10 times: 1 0 1 1 1 0 1 1 0 1 Average=0.7

3. More flips n=100: 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 average=0.51 n=10,000, we would expect to get 5000 heads and 5000 tails. average=0.508

4. What is the value for expected long run average? Conjecture: ½ ½ probability to get 0, ½ probability to get 1 (½) (0)+ (½) (1)=1/2

5. Roll a die With equal probabilities 1/6 x123456x123456 p(x)1/61/61/61/61/61/6 Toss 6000 times  about 1,000 of each x-value.

6. Roll some more Some simulations: Roll n=10 times: 6 5 6 4 6 3 4 1 2 2 Average=3.9 x<-round(runif(10)*6+0.5) n=100 1000 10,000 100,000 Average 3.56 3.527 3.5008 3.49386 Average 3.49949  3.5

7. For numerical outcomes Get x with probability P(x) Values x 1 x 2 … x k Prob p 1 p 2 … p k P(X 1 ) P(X 2 ) … P(X k ) Mathematical expectation of X is given by E=E(x)= x 1 p 1 +x 2 p 2 +…+ x k p k = x 1 p(x 1 )+x 2 p(x 2 )+…+ x k p(x k )

8. Raffle ticket x$0$100 p(x)199/2001/200 This is the population mean for the population of possible ticket prizes. 0 0 0 100 1 out of every 200 tickets

9. Example 7.1 Toss a fair coin until a head or quit at 3 tosses Expected tosses needed? X P(x) 1 ½ H 2 ¼=(½) (½) TH 3 ¼ TTH, TTT E(X)=(1)(½)+(2)(¼)+(3)(¼)=1.75 If we repeated this experiment over and over, we would average 1.75 tosses.

10. Example 7.3 Gambling: A and B roll two dice. If A’s number is larger, A wins dollars for the amount he got on the top of the die, otherwise, A loses $3. Expected gain of A?

11. Solution x P(x) -3 21/36 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 E=7/36

12. Example X=# of birds fledged from a nest xp(x) 00.2 10.2 20.4 30.2 1.0 What is the expected value of x? On average, how many birds are fledged per nest?

13. 1 22 3 0 20%0 20%1 40%2 20%3  =1.6