1 / 13

Chapter 7 Expectation

Chapter 7 Expectation. 7.1 Mathematical expectation. 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. More flips. n=100:

iola-sosa
Download Presentation

Chapter 7 Expectation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 x 1 2 3 4 5 6 p(x) 1/6 1/6 1/6 1/6 1/6 1/6 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 x1 x2 … xk Prob p1 p2 … pk P(X1) P(X2) … P(Xk) Mathematical expectation of X is given by E=E(x)= x1 p1+x2p2+…+ xkpk = x1 p(x1)+x2p(x2)+…+ xkp(xk)

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

  9. Example 7.1 • Toss a fair coin until a head or quit at 3 tosses Expected tosses needed? X P(x) • ½ H • ¼=(½) (½) TH • 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) 0 0.2 1 0.2 2 0.4 3 0.2 1.0 What is the expected value of x? On average, how many birds are fledged per nest?

  13. 20% 0 20% 1 40% 2 20% 3 m=1.6 1 3 2 2 0

More Related