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Unformatted text preview: MA2216/ST2131 Probability Notes 10 Properties of Expectation and Inequalities § 1. Summary of Basic Properties. Let us first review some elementary properties of mathematical expec tation, and then develop and exploit additional properties and useful calculations. (Recall that mathematical expectation had been defined in § 4 of Notes 3 in a notquite rigorous way.) 1. To begin with, take note that the expected value of a discrete r.v. X can be defined if X x  x  f X ( x ) < ∞ , i.e., the series X x  x  f X ( x ) converges. In such a case, the expected value (or mean) μ X of X is given by μ X = EE [ X ] = X x xf X ( x ) . Analytically speaking , the series X x xf X ( x ) has to converge abso lutely , i.e., X x  x  f X ( x ) < ∞ , so that the expected value of X may be defined. For a continuous r.v. X , its expected value exists if Z IR  x  f X ( x ) dx < ∞ , and in such a case, μ X = EE [ X ] = Z IR xf X ( x ) dx. In short, EE [ X ] exists if EE [  X  ] < ∞ . In a similar fashion, we say EE [ g ( X )] is defined if EE [  g ( X )  ] < ∞ . 1 2. The above is the univariate case. For the multivariate case, (a) If X and Y are jointly discrete with joint p.d.f. f X,Y , then EE [ g ( X,Y )] = X x X y g ( x,y ) f X,Y ( x,y ) , provided that X x X y  g ( x,y )  f X,Y ( x,y ) < ∞ . (b) If X and Y are jointly continuous with joint p.d.f. f X,Y , then EE [ g ( X,Y )] = Z ∞∞ Z ∞∞ g ( x,y ) f X,Y ( x,y ) dxdy, provided that Z ∞∞ Z ∞∞  g ( x,y )  f X,Y ( x,y ) dxdy < ∞ . 3. Note. Not every r.v. has finite mean. We give two examples for refer ence. (a) Consider a positive integervalued r.v. X having p.d.f. f X given by f X ( k ) = 1 k ( k + 1) , k = 1 , 2 ,... (This is a genuine probability density function, for it is nonnegative everywhere and its total sum over positive integers is 1.) But, ∞ X k =1 k f X ( k ) = ∞ X k =1 1 k + 1 = ∞ , and hence X does not have finite mean, (or its mean is not defined). (b) Let X have the p.d.f. f given by f ( x ) = 1 π (1 + x 2 ) ,∞ < x < ∞ . (This is also known as a Cauchy distribution.) Then, X does not have finite expectation. For Z ∞∞  x  1 π (1 + x 2 ) dx = 2 π Z ∞ x 1 + x 2 dx = 2 π lim c →∞ Z c x 1 + x 2 dx = 1 π lim c →∞ log (1 + x 2 ) fl fl c = ∞ . 2 Next, some basic properties of expectation. 4. Linearity. If EE  X i  < ∞ for i = 1 , 2 , ··· ,n , then EE ˆ n X i =1 X i ! = n X i =1 EEX i . More generally, for r.v.’s X 1 ,X 2 ,...,X n with finite means, we have EE " n X k =1 α k X k + β # = n X k =1 α k EE [ X k ] + β, where α k ’s and β are constants. For the multivariate case, EE [ g ( X,Y ) + h ( X,Y )] = EE [ g ( X,Y )] + EE [ h ( X,Y )] ....
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