elemprob-fall2010-page6

elemprob-fall2010-page6 - cases later on) that this...

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3 Expected value Expected value and expectation mean the same thing. For a discrete random variable, it is defined by E X = X x P ( X = x ) . The sum is over all x for which P ( X = x ) 6 = 0. For example, if we roll a die and X is the value that shows, E X = 1 · 1 6 + 2 · 1 6 + ··· + 6 · 1 6 = 35 6 = 3 . 5 . If we do an experiment over and over and X 1 ,X 2 ,... are our observations, then X 1 + ··· + X n n is the average value, and it is a theorem (that we will prove in some special
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Unformatted text preview: cases later on) that this converges to E X . When Ω is countable, there is an equivalent definition of expectation which is also useful. Proposition 3.1 E X = X ω ∈ Ω X ( ω ) P ( ω ) . Proof. E X = X x P ( X = x ) = X x x X { ω : X ( ω )= x } P ( ω ) = X x X { ω : X ( ω )= x } X ( ω ) P ( ω ) = X ω X ( ω ) P ( ω ) . Let f : R → R . 6...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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