Mathematical Expectation

31.10.2013Dr.  Bilge  Eriş

“Mathematical  Expectation”• arose in connection with games of chance• Amount the player stands to win*probability

that he will win– one of the 10.000 tickets with a prize of $4800– one of the 10.000 tickets with a prize of $1200– one of the 10.000 tickets with a prize of $400

• On average win;

– Sum of the products obtained by multiplying each amount by the corresponding probability

• if there is a second prize worth 1200$ and third worth 400$– On average 4800+1200+400/10.000=0.64$ per

ticket

4.2 THE EXPECTED VALUE OF A RANDOM VARIABLE

• The amount won: Random variable• Sum of the products: Mathematical

expectation• Refer mathematical expectation of a random

variable as expected value

We may be expected in expected values of random variables related with x

Calculating expected values from other known expectations

Possible to extend mathematical expectation

4.3 MOMENTS• Mathematical expectations called moments of

a random variable

When r=1, we have

Other special moments

4.4  CHEBYSHEV’S  THEOREM

• P(X will take on value within 2 sd. of the mean)=3/4 at least

• P(X will take on value within 3 sd. of the mean)=8/9 at least

• Only when the distribution is known, we can calculate the exact probability

4.5 MOMENT GENERATING FUNCTIONS

• Alternative procedure to calculate moments

Basic theorems

4.6 PRODUCT MOMENTS

• Indepence implies zero covariance but zero covariance does NOT imply independence!

Generalization of product moments

4.7 MOMENTS OF LINEAR COMBINATIONS OF RANDOM VARIABLES

• Derive mean, variance of linear combinations

4.8 CONDITIONAL EXPECTATION

• Likewise the conditional distributions