note 10 - MA2216/ST2131 Probability Notes 10 Properties of...

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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 not-quite 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 integer-valued 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 non-negative 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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note 10 - MA2216/ST2131 Probability Notes 10 Properties of...

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