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MIT6_042JF10_rec22_sol

MIT6_042JF10_rec22_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science December 3, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 22 1 Expected Value Rule for Functions of Random Vari- ables In lecture, we have computed the expectation of a function of a random variable without explicitly discussing the general rule. For example, yesterday we saw that the expectation of the square of the roll of a die is not equal to the square of the expectation of the roll. That is, if R is the outcome of a single roll of a die. Then Ex ( R 2 ) = Ex ( R ) 2 . We will now explicitly present the rule for computing the expection of a function of a random variable R . Rule (Expected Value for the Function of a Random Variable) . Let R be a random variable, and let f ( R ) be a function of R . Then, the expected value of the random variable f ( R ) is given by E [ f ( R )] = f ( x ) Pr { R = x } · x ∈ Range ( R ) 2 Properties of Variance In this problem we will study some properties of the variance and the standard deviation of random variables. a. Show that for any random variable R , Var [ R ] = E [ R 2 ] − E 2 [ R ]. Solution. Let µ = E [ R ]. Then Var [ R ] = E ( R − E [ R ]) 2 (Definition of variance) = E ( R − µ ) 2 (def. of µ ) = E R 2 − 2 µR + µ 2 = E R 2 − 2 µ E [ R ] + µ 2 (linearity of expectation) = E R 2 − 2 µ 2 + µ 2 (def. of µ ) = E R 2 − µ 2 = E R 2 − E 2 [ R...
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MIT6_042JF10_rec22_sol - 6.042/18.062J Mathematics for...

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