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Characteristic Functions
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If we let t iv, where i is the imaginary unit, in the moment generating function we obtain an important function called the characteristic function. We denote this by fX (v) It follows that fX(v)
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a eivx f(x)
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(discrete variable)
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(continuous variable)
(37)
Since u eivx u 1, the series and the integral always converge absolutely. The corresponding results (31) and (32) become fX(v) 1 imv mr 2 v2 2! c irmrr vr r! c (38)
where
mr r
( 1)rir
dr f (v) 2 dvr X v
(39)
When no confusion can result, we often write f(v) instead of fX(v). Theorems for characteristic functions corresponding to Theorems 3-8, 3-9, and 3-10 are as follows. Theorem 3-11 If fX(v) is the characteristic function of the random variable X and a and b (b 2 0) are constants, then the characteristic function of (X a) > b is f(X Theorem 3-12
a)>b(v)
If X and Y are independent random variables having characteristic functions fX (v) and fY (v), respectively, then fX Y (v) fX (v) fY (v) (41)
v eaiv>bfX b
(40)
More generally, the characteristic function of a sum of independent random variables is equal to the product of their characteristic functions. Theorem 3-13 (Uniqueness Theorem) Suppose that X and Y are random variables having characteristic functions fX (v) and fY (v), respectively. Then X and Y have the same probability distribution if and only if fX (v) fY (v) identically.
CHAPTER 3 Mathematical Expectation
An important reason for introducing the characteristic function is that (37) represents the Fourier transform of the density function f (x). From the theory of Fourier transforms, we can easily determine the density function from the characteristic function. In fact, f (x) 1 ` e 2p 3 `
ivx f (v) dv X
(42)
which is often called an inversion formula, or inverse Fourier transform. In a similar manner we can show in the discrete case that the probability function f(x) can be obtained from (36) by use of Fourier series, which is the analog of the Fourier integral for the discrete case. See Problem 3.39. Another reason for using the characteristic function is that it always exists whereas the moment generating function may not exist.
Variance for Joint Distributions. Covariance
The results given above for one variable can be extended to two or more variables. For example, if X and Y are two continuous random variables having joint density function f(x, y), the means, or expectations, of X and Y are
` ` ` ` `
mX and the variances are
E(X)
3 ` 3 `xf (x, y) dx dy,
E(Y)
3 `yf (x, y) dx dy
(43)
s2 X s2 Y
E[(X E[(Y
mX )2] mY)2]
3 `(x
mX)2 f(x, y) dx dy (44) mY)2 f(x, y) dx dy
3 `( y
Note that the marginal density functions of X and Y are not directly involved in (43) and (44). Another quantity that arises in the case of two variables X and Y is the covariance defined by sXY Cov (X, Y) E[(X mX)(Y mY)] (45)
In terms of the joint density function f (x, y), we have
3 `(x `
mX)(y
mY) f(x, y) dx dy
(46)
Similar remarks can be made for two discrete random variables. In such cases (43) and (46) are replaced by mX a a xf(x, y)
mY mX)( y
a a yf(x, y)
(47) (48)
a a (x
mY) f(x, y)
where the sums are taken over all the discrete values of X and Y. The following are some important theorems on covariance. Theorem 3-14 Theorem 3-15 sXY E(XY ) E(X)E(Y ) E(XY ) mXmY (49)
If X and Y are independent random variables, then sXY Cov (X, Y ) Var (Y ) s2 Y 0 2 Cov (X, Y ) 2sXY (50) (51) (52) (53)
Theorem 3-16 or Theorem 3-17
Var (X
Var (X) s2 Y X s2 X ZsXY Z
sX sY
CHAPTER 3 Mathematical Expectation
The converse of Theorem 3-15 is not necessarily true. If X and Y are independent, Theorem 3-16 reduces to Theorem 3-7.
Correlation Coefficient
If X and Y are independent, then Cov(X, Y) sXY for example, when X Y, then Cov(X, Y) sXY of the variables X and Y given by r 0. On the other hand, if X and Y are completely dependent, sX sY. From this we are led to a measure of the dependence sXY sX sY
(54)
We call r the correlation coefficient, or coefficient of correlation. From Theorem 3-17 we see that 1 r 1. In the case where r 0 (i.e., the covariance is zero), we call the variables X and Y uncorrelated. In such cases, however, the variables may or may not be independent. Further discussion of correlation cases will be given in 8.
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