# Lect6 - 6 Mean Variance Moments and Characteristic Functions For a r.v X its p.d.f f X x represents complete information about it and for any Borel

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1 6. Mean, Variance, Moments and Characteristic Functions For a r.v X , its p.d.f represents complete information about it, and for any Borel set B on the x -axis Note that represents very detailed information, and quite often it is desirable to characterize the r.v in terms of its average behavior. In this context, we will introduce two parameters - mean and variance - that are universally used to represent the overall properties of the r.v and its p.d.f. () = B X dx x f B X P . ) ( ) ( ξ (6-1) ) ( x f X ) ( x f X PILLAI

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2 Mean or the Expected Value of a r.v X is defined as If X is a discrete-type r.v, then using (3-25) we get Mean represents the average (mean) value of the r.v in a very large number of trials. For example if then using (3-31) , is the midpoint of the interval ( a , b ). + = = = . ) ( ) ( dx x f x X E X X X η (6-2) . ) ( ) ( ) ( ) ( 1 = = = = = = = i i i i i i i i i i i i i X x X P x p x dx x x p x dx x x p x X E X ± ±² ± ±³ ´ δ δ η (6-3) ), , ( b a U X (6-4) + = = = = b a b a b a a b a b x a b dx a b x X E 2 ) ( 2 2 1 ) ( 2 2 2 PILLAI
3 On the other hand if X is exponential with parameter as in (3-32), then implying that the parameter in (3-32) represents the mean value of the exponential r.v. Similarly if X is Poisson with parameter as in (3-43), using (6-3), we get Thus the parameter in (3-43) also represents the mean of the Poisson r.v. = = = 0 / 0 , ) ( λ λ λ λ dy ye dx e x X E y x (6-5) λ λ λ . ! )! 1 ( ! ! ) ( ) ( 0 1 1 0 0 λ λ λ λ λ λ λ λ λ λ λ λ λ = = = = = = = = = = = = = e e i e k e k k e k ke k X kP X E i i k k k k k k k (6-6) λ PILLAI

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4 In a similar manner, if X is binomial as in (3-42), then its mean is given by Thus np represents the mean of the binomial r.v in (3-42). For the normal r.v in (3-29), . ) ( ! )! 1 ( )! 1 ( )! 1 ( )! ( ! ! )! ( ! ) ( ) ( 1 1 1 0 1 1 0 0 np q p np q p i i n n np q p k k n n q p k k n n k q p k n k k X kP X E n i n i n i k n k n k k n k n k k n k n k n k = + = = = = = = = = = = = = (6-7) . 2 1 2 1 ) ( 2 1 2 1 ) ( 1 2 / 2 0 2 / 2 2 / 2 2 / ) ( 2 2 2 2 2 2 2 2 2 µ πσ µ πσ µ πσ πσ σ σ σ σ µ = + = + = = + + + + ± ± ±² ± ± ±³ ´ ± ± ´ dy e dy ye dy e y dx xe X E y y y x (6-8) PILLAI
5 Thus the first parameter in is infact the mean of the Gaussian r.v X . Given suppose defines a new r.v with p.d.f Then from the previous discussion, the new r.v Y has a mean given by (see (6-2)) From (6-9), it appears that to determine we need to determine However this is not the case if only is the quantity of interest. Recall that for any y , where represent the multiple solutions of the equation But(6-10) can be rewritten as ) , ( 2 σ µ N X ), ( x f X X ) ( X g Y = ). ( y f Y Y µ + = = . ) ( ) ( dy y f y Y E Y Y µ (6-9) ), ( Y E ). ( y f Y ) ( Y E 0 > y () ( ) , + < = + < i i i i x x X x P y y Y y P (6-10) i x ). ( i x g y = , ) ( ) ( i i i X Y x x f y y f = (6-11) PILLAI

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6 where the terms form nonoverlapping intervals.
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## This note was uploaded on 10/01/2009 for the course ECE 2521 taught by Professor Jacobs during the Fall '09 term at Pittsburgh.

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Lect6 - 6 Mean Variance Moments and Characteristic Functions For a r.v X its p.d.f f X x represents complete information about it and for any Borel

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