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

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

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