1Welcome Back Our Friend: Expectation•Recall expectation for discrete random variable:•Analogously for a continuous random variable:•Note: If X always between aand bthen so is E[X]More formally:xxpxXE)(][dxxfxXE)(][bXEabXaP][1)(thenifGeneralizing Expectation•Let g(X, Y) be real-valued function of two variables•Let X and Y be discrete jointly distributed RVs:•Analogously for continuous random variables:yxYXyxpyxgYXgE),(),()],([,dydxyxfyxgYXgEyxYX),(),()],([,Expected Values of Sums•Let g(X, Y) = X + Y. Compute E[g(X, Y)] = E[X + Y]•Generalized:Holds regardless of dependency between Xi’sdydxyxfyxYXEyxYX),()(][,dydxyxfydxdyyxfxyxYXxyYX),(),(,,dyyfydxxfxyYxX)()(][YEXEniiniiXEXE11][Tie Me Up! : Bounding Expectation•If random variable X ≥ athen E[X] ≥ aOften useful in cases where a= 0But, E[X] ≥ adoes notimply X ≥ afor all X = xoE.g., X is equally likely to take on values -1 or 3. E[X] = 1.•If random variables X ≥ Y then E[X] ≥ E[Y]X ≥ Y X – Y ≥ 0 E[X – Y] ≥ 0Note: E[X –Y] = E[X] + E[-Y] = E[X] –E[Y]Substituting: E[X] – E[Y] ≥ 0 E[X] ≥ E[Y]But, E[X] ≥ E[Y] does notimply X ≥ Y for all X = x, Y = y][1)(XEaXaPthenif
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