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 x1, x2, x3, ... with corresponding probabilities p(x1), p(x2), p(x3),..... The mean or expected value of X is defined by E(X) = sum xk p(xk). 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)

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 X=3 X=5 X=-4 P(X=1) = P({1}) =1/6 P(X=1) = P({3}) =1/6 P(X=1) = P({5}) =1/6 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

Ex. 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 x1, x2, x3, ... with corresponding probabilities p(x1), p(x2), p(x3),..... For any function g, the mean or expected value of g(X) is defined by E(g(X)) = sum g(xk) p(xk). Ex. Roll a fair die. Let X = number of dots on the side that comes up. Calculate E(X2). E(X2) = sum_{i=1}^{6} i2 p(i) = 12 p(1) + 22 p(2) + 32 p(3) + 42 p(4) + 52 p(5) + 62 p(6) = 1/6*(1+4+9+16+25+36) = 91/6 E(X) is the expected value or 1st moment of X. E(Xn)