Chapter 7 Expectation

7.1 Mathematical expectation

7.1 Mathematical Expectation

Mathematical expectation =expected long run average

Simulation 1: toss a fair coin H1, T0. n=10 times: 1 0 1 1 1 0 1 1 0 1Average=0.7

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 0average=0.51

n=10,000, we would expect to get 5000 heads and 5000 tails.

average=0.508

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

Roll a die

With equal probabilities 1/6 x 1 2 3 4 5 6p(x) 1/6 1/6 1/6 1/6 1/6 1/6

Toss 6000 times about 1,000 of each x-value.

1000*(1) 1000*(2) 1000*(3) 1000*(4) 1000*(5) 1000*(6)

1000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 1000

(1) (2) (3) (4) (5) (6)6000 6000 6000 6000 6000 6000

1 1 1 1 1 1(1) (2) (3) (4) (5) (6)

6 6 6 6 6 6(

Mean

xp

) (Not book's notation exactly)

= weighted average of x, weighted by the probability of each possible x value

1=3 3.5

2

x

Roll some moreSome simulations:

Roll n=10 times: 6 5 6 4 6 3 4 1 2 2Average=3.9x<-round(runif(10)*6+0.5)

n=100 1000 10,000 100,000Average 3.56 3.527 3.5008 3.49386

Average 3.49949 3.5

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 byE=E(x)= x1 p1+x2p2+…+ xkpk

= x1 p(x1)+x2p(x2)+…+ xkp(xk)

Raffle ticketx $0 $100p(x) 199/200 1/200

This is the population mean for the population of possible ticket prizes.

199 1( ) ( ) 0*( ) 100*( ) 0.50

200 200E x xp x

0

0

0

100

1 out of every 200 tickets

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) TH3 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.

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?

Solution

x P(x)-3 21/362 1/36 3 2/364 3/365 4/366 5/36E=7/36

Example

X=# of birds fledged from a nest

x p(x)0 0.21 0.22 0.43 0.2

1.0What is the expected value of x?On average, how many birds are fledged per nest?

( ) ( )

0*0.2 1*0.2 2*0.4 3*0.2

1.6

E x xp x

1

2 2

3

0

20% 0

20% 1

40% 2

20% 3

=1.6