elemprob-page15

elemprob-page15 - If x[k/2n(k 1/2n then x diers from k/2n...

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If x [ k/ 2 n , ( k + 1) / 2 n ), then x differs from k/ 2 n by at most 1 / 2 n . So the last integral differs from ( k +1) / 2 n k/ 2 n xf ( x ) dx by at most (1 / 2 n ) P ( k/ 2 n X < ( k + 1) / 2 n ) 1 / 2 n , which goes to 0 as n → ∞ . On the other hand, ( k +1) / 2 n k/ 2 n xf ( x ) dx = M 0 xf ( x ) dx, which is how we defined the expectation of X . We will not prove the following, but it is an interesting exercise: if X m is any sequence of discrete random variables that increase up to X , then lim m →∞ E X m will have the same value E X . To show linearity, if X and Y are bounded positive random variables, then take X m discrete increasing up to X and Y m discrete increasing up to Y . Then X m + Y m is discrete and increases up to X + Y , so we have E ( X + Y ) = lim m →∞ E ( X m + Y m ) = lim m →∞ E X m + lim m →∞ E Y m = E X + E Y. If X is not bounded or not necessarily positive, we have a similar definition; we will not do the details. This second definition of expectation is mostly useful for theoretical purposes and much less so for calculations. Similarly to the discrete case, we have
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