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ssrs 2008 r2 barcode font Characteristic Functions in Software
Characteristic Functions QR Code Reader In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Code JIS X 0510 Printer In None Using Barcode creator for Software Control to generate, create Denso QR Bar Code image in Software applications. 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) Scanning Quick Response Code In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Create Denso QR Bar Code In Visual C#.NET Using Barcode creator for Visual Studio .NET Control to generate, create QR Code image in .NET applications. MX (iv) Paint QR In Visual Studio .NET Using Barcode creator for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. QR Code Creation In VS .NET Using Barcode creation for .NET framework Control to generate, create QRCode image in .NET framework applications. E(eivX) Generate QR Code ISO/IEC18004 In Visual Basic .NET Using Barcode creator for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. EAN / UCC  13 Generator In None Using Barcode creation for Software Control to generate, create EAN128 image in Software applications. (35) Code 39 Extended Creation In None Using Barcode printer for Software Control to generate, create USS Code 39 image in Software applications. Barcode Encoder In None Using Barcode generator for Software Control to generate, create bar code image in Software applications. a eivx f(x) DataMatrix Creator In None Using Barcode drawer for Software Control to generate, create Data Matrix image in Software applications. Print UCC  12 In None Using Barcode creator for Software Control to generate, create UPCA image in Software applications. (discrete variable) Generate ITF14 In None Using Barcode maker for Software Control to generate, create UCC  14 image in Software applications. Make Bar Code In VB.NET Using Barcode printer for .NET Control to generate, create bar code image in .NET framework applications. (36) Bar Code Drawer In None Using Barcode generator for Font Control to generate, create bar code image in Font applications. Creating GS1 128 In VS .NET Using Barcode creator for .NET Control to generate, create EAN / UCC  13 image in .NET framework applications. fX(v) Decoding Data Matrix 2d Barcode In C# Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Matrix Barcode Drawer In .NET Using Barcode printer for VS .NET Control to generate, create Matrix Barcode image in VS .NET applications. ivx 3 `e f(x) dx
GTIN  13 Maker In None Using Barcode encoder for Font Control to generate, create EAN13 image in Font applications. UPCA Recognizer In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. (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 38, 39, and 310 are as follows. Theorem 311 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 312 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 313 (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 314 Theorem 315 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 316 or Theorem 317 Var (X
Var (X) s2 Y X s2 X ZsXY Z
sX sY
CHAPTER 3 Mathematical Expectation
The converse of Theorem 315 is not necessarily true. If X and Y are independent, Theorem 316 reduces to Theorem 37. 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 317 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.

