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x, Y QR Code 2d Barcode Reader In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encode QR Code ISO/IEC18004 In None Using Barcode maker for Software Control to generate, create Quick Response Code image in Software applications. f(x, y) Recognize QR Code JIS X 0510 In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Drawing QR Code 2d Barcode In C# Using Barcode generation for .NET Control to generate, create Denso QR Bar Code image in .NET framework applications. (13) Denso QR Bar Code Generator In .NET Framework Using Barcode generator for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QR Code Printer In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create QR image in .NET framework applications. 2. a a f (x, y) QR Code 2d Barcode Creation In VB.NET Using Barcode creator for .NET Control to generate, create QRCode image in VS .NET applications. Create Code 128 Code Set B In None Using Barcode creation for Software Control to generate, create Code 128 Code Set A image in Software applications. i.e., the sum over all values of x and y is 1. Suppose that X can assume any one of m values x1, x2, . . . , xm and Y can assume any one of n values y1, y2, . . . , yn. Then the probability of the event that X xj and Y yk is given by P(X xj, Y yk) f(xj, yk) (14) Print GTIN  128 In None Using Barcode creation for Software Control to generate, create GTIN  128 image in Software applications. UPCA Printer In None Using Barcode printer for Software Control to generate, create UPC Code image in Software applications. A joint probability function for X and Y can be represented by a joint probability table as in Table 23. The probability that X xj is obtained by adding all entries in the row corresponding to xi and is given by Barcode Maker In None Using Barcode creation for Software Control to generate, create barcode image in Software applications. Data Matrix ECC200 Creation In None Using Barcode creation for Software Control to generate, create Data Matrix image in Software applications. f1(xj) USPS POSTNET Barcode Encoder In None Using Barcode printer for Software Control to generate, create USPS POSTNET Barcode image in Software applications. Making Code 128 Code Set C In ObjectiveC Using Barcode creator for iPhone Control to generate, create Code 128 Code Set A image in iPhone applications. a f (xj, yk) Barcode Creation In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. Data Matrix ECC200 Generation In Visual Studio .NET Using Barcode printer for Reporting Service Control to generate, create Data Matrix image in Reporting Service applications. (15) EAN13 Drawer In Java Using Barcode creator for Eclipse BIRT Control to generate, create GS1  13 image in Eclipse BIRT applications. EAN13 Recognizer In Visual C#.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Table 23 Y X x1 x2 ( xm Totals S y1 f (x1, y1) f (x2, y1) ( f (xm, y1 ) f2 (y1 ) y2 f (x1, y2) f (x2, y2) ( f (xm, y2 ) f2 (y2 ) c c c c c yn f(x1, yn ) f(x2, yn ) ( f(xm, yn) f2 (yn) Totals T f1 (x1) f1 (x2) ( f1 (xm) 1 d Grand Total Generate Code 3 Of 9 In Visual Basic .NET Using Barcode creation for VS .NET Control to generate, create Code 39 Extended image in VS .NET applications. Universal Product Code Version A Printer In VS .NET Using Barcode creation for Reporting Service Control to generate, create UPC A image in Reporting Service applications. For j 1, 2, . . . , m, these are indicated by the entry totals in the extreme righthand column or margin of Table 23. Similarly the probability that Y yk is obtained by adding all entries in the column corresponding to yk and is given by f2(yk ) a f (xj, yk ) (16) For k 1, 2, . . . , n, these are indicated by the entry totals in the bottom row or margin of Table 23. Because the probabilities (15) and (16) are obtained from the margins of the table, we often refer to f1(xj) and f2(yk) [or simply f1(x) and f2(y)] as the marginal probability functions of X and Y, respectively. CHAPTER 2 Random Variables and Probability Distributions
It should also be noted that
a f1 (xj) 1 a f2 (yk) (17) which can be written
a a f (xj, yk) j 1k 1
(18) This is simply the statement that the total probability of all entries is 1. The grand total of 1 is indicated in the lower righthand corner of the table. The joint distribution function of X and Y is defined by F(x, y) P(X x, Y y) a a f (u, v) y. (19) u xv y
In Table 23, F(x, y) is the sum of all entries for which xj
x and yk
2. CONTINUOUS CASE. The case where both variables are continuous is obtained easily by analogy with the discrete case on replacing sums by integrals. Thus the joint probability function for the random variables X and Y (or, as it is more commonly called, the joint density function of X and Y ) is defined by 1. f(x, y) ` ` `
2. 3 f (x, y) dx dy
Graphically z f(x, y) represents a surface, called the probability surface, as indicated in Fig. 24. The total volume bounded by this surface and the xy plane is equal to 1 in accordance with Property 2 above. The probability that X lies between a and b while Y lies between c and d is given graphically by the shaded volume of Fig. 24 and mathematically by b, c
3 a y
f (x, y) dx dy
(20) Fig. 24 More generally, if A represents any event, there will be a region 5A of the xy plane that corresponds to it. In such case we can find the probability of A by performing the integration over 5A, i.e., P(A) 33 f (x, y) dx dy (21) The joint distribution function of X and Y in this case is defined by
x y `
F(x, y) x, Y
f (u, v) du dv
(22) CHAPTER 2 Random Variables and Probability Distributions
It follows in analogy with (11), page 38, that '2F 'x 'y f (x, y) (23) i.e., the density function is obtained by differentiating the distribution function with respect to x and y. From (22) we obtain F1(x) 3 ` v
f (u, v) du dv f (u, v) du dv
(24) F2( y) 3 ` v
(25) We call (24) and (25) the marginal distribution functions, or simply the distribution functions, of X and Y, respectively. The derivatives of (24) and (25) with respect to x and y are then called the marginal density functions, or simply the density functions, of X and Y and are given by f1(x)

