# chapter4 - Expected Value The expected value of a random...

This preview shows pages 1–5. Sign up to view the full content.

Expected Value The expected value of a random variable indicates its weighted average. Ex. How many heads would you expect if you flipped a coin twice? X = number of heads = {0,1,2} p(0)=1/4, p(1)=1/2, p(2)=1/4 Weighted average = 0*1/4 + 1*1/2 + 2*1/4 = 1 Draw PDF Definition: Let X be a random variable assuming the values x 1 , x 2 , x 3 , . .. with corresponding probabilities p(x 1 ), p(x 2 ), p(x 3 ),. .... The mean or expected value of X is defined by E(X) = sum x k p(x k ). Interpretations: (i) The expected value measures the center of the probability distribution - center of mass. (ii) Long term frequency (law of large numbers… we’ll get to this soon)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Expectations can be used to describe the potential gains and losses from games. Ex. Roll a die. If the side that comes up is odd, you win the \$ equivalent of that side. If it is even, you lose \$4. Let X = your earnings X=1 P(X=1) = P({1}) =1/6 X=3 P(X=1) = P({3}) =1/6 X=5 P(X=1) = P({5}) =1/6 X=-4 P(X=1) = P({2,4,6}) =3/6 E(X) = 1*1/6 + 3*1/6 + 5*1/6 + (-4)*1/2 = 1/6 + 3/6 +5/6 – 2= -1/2 Lottery – You pick 3 different numbers between 1 and 12. If you pick all the numbers correctly you win \$100. What are your expected earnings if it costs \$1 to play? Let X = your earnings X = 100-1 = 99 X = -1 P(X=99) = 1/(12 3) = 1/220 P(X=-1) = 1-1/220 = 219/220 E(X) = 100*1/220 + (-1)*219/220 = -119/220 = -0.54
Expectation of a function of a random variable Let X be a random variable assuming the values x 1 , x 2 , x 3 , . .. with corresponding probabilities p(x 1 ), p(x 2 ), p(x 3 ),. .... For any function g, the mean or expected value of g(X) is defined by E(g(X)) = sum g(x k ) p(x k ). Ex. Roll a fair die. Let X = number of dots on the side that comes up. Calculate E(X 2 ). E(X 2 ) = sum_{i=1}^{6} i 2 p(i) = 1 2 p(1) + 2 2 p(2) + 3 2 p(3) + 4 2 p(4) + 5 2 p(5) + 6 2 p(6) = 1/6*(1+4+9+16+25+36) = 91/6 E(X) is the expected value or 1 st moment of X. E(X n ) is called the nth moment of X. Calculate E(sqrt(X)) = sum_{i=1}^{6} sqrt(i) p(i) Calculate E(e X ) = sum_{i=1}^{6} e i p(i) (Do at home) An indicator variable for the event A is defined as the random variable that takes on the value 1 when event A happens and 0 otherwise. I A = 1 if A occurs 0 if A C occurs P( I A =1) = P(A) and P( I A =0) = P(A C ) The expectation of this indicator (noted I A ) is E( I A )=1*P(A) + 0*P(A C ) =P(A). One-to-one correspondence between expectations and probabilities. If a and b are constants, then E(aX+b) = aE(X) + b Proof: E(aX+b) = sum [(ax k +b) p(x k )] = a sum{x k p(x k )} + b sum{p(x k )} = aE(X) + b

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Variance We often seek to summarize the essential properties of a random variable in as simple terms as possible. The mean is one such property. Let X = 0 with probability 1 Let Y = -2 with prob. 1/3 -1 with prob. 1/6 1 with prob. 1/6 2 with prob. 1/3 Both X and Y have the same expected value, but are quite different in other respects. One such respect is in their spread. We would like a measure of spread. Definition: If X is a random variable with mean E(X), then the variance of X, denoted by Var(X), is defined by Var(X) = E((X-E(X)) 2 ).
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 02/13/2012 for the course ECON 101 taught by Professor Teerana during the Spring '11 term at Thammasat University.

### Page1 / 21

chapter4 - Expected Value The expected value of a random...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online