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# lect6a - 1 6 Mean Variance Moments and Characteristic...

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Unformatted text preview: 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 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 ¡ ¡ ¢ ¡ ¡ £ ¤ 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 (3-45), using (6-3), we get Thus the parameter in (3-45) also represents the mean of the Poisson r.v. ∫ ∫ ∞ − − ∞ = = = / , ) ( λ λ λ λ dy ye dx e x X E y x (6-5) λ λ λ . ! )! 1 ( ! ! ) ( ) ( 1 1 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 4 In a similar manner, if X is binomial as in (3-44), then its mean is given by Thus np represents the mean of the binomial r.v in (3-44). For the normal r.v in (3-29), . ) ( ! )! 1 ( )! 1 ( )! 1 ( )! ( ! ! )! ( ! ) ( ) ( 1 1 1 1 1 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 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 = ⋅ + = + = = ∫ ∫ ∫ ∫ ∞ + ∞ − − ∞ + ∞ − − ∞ + ∞ − − ∞ + ∞ − − − ¡ ¡ ¡ ¢ ¡ ¡ ¡ £ ¤ ¡ ¡ ¢ ¡ ¡ £ ¤ dy e dy ye dy e y dx xe X E y y y x (6-8) PILLAI 5 hus the first parameter in ∼ is infact the mean of the Gaussian r.v X . Given ∼ suppose defines a ew r.v with p.d.f Then from the previous discussion, the new r.v Y has a mean given by (see (6-2)) rom (6-9), it appears that to determine we need to etermine However this is not the case if only is the quantity of interest. Recall that for any y , here represent the multiple solutions of the equation...
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lect6a - 1 6 Mean Variance Moments and Characteristic...

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