LINEAR COMBINATIONS OF RANDOM VARIABLES in .NET Generation QR Code in .NET LINEAR COMBINATIONS OF RANDOM VARIABLES LINEAR COMBINATIONS OF RANDOM VARIABLESGenerate qr for .netgenerate, create qr code none with .net projectsA random variable is sometimes de ned as a function of one or more random variables. The CD material presents methods to determine the distributions of general functions of random variables. Furthermore, moment-generating functions are introduced on the CD.net Vs 2010 qr codes reader for .netUsing Barcode decoder for visual .net Control to read, scan read, scan image in visual .net applications.5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES Make bar code on .netuse vs .net bar code maker toinsert bar code with .netand used to determine the distribution of a sum of random variables. In this section, results for linear functions are highlighted because of their importance in the remainder of the book. References are made to the CD material as needed. For example, if the random variables X1 and X2 denote the length and width, respectively, of a manufactured part, Y 2X1 2X2 is a random variable that represents the perimeter of the part. As another example, recall that the negative binomial random variable was represented as the sum of several geometric random variables. In this section, we develop results for random variables that are linear combinations of random variables.Integrate barcode on .netgenerate, create barcode none in .net projectsDe nition Given random variables X1, X2, p , Xp and constants c1, c2, p , cp, Y c1 X1 c2 X2 p cp Xp (5-36)Control denso qr bar code image in c#using barcode development for .net framework control to generate, create qr-codes image in .net framework applications.is a linear combination of X1, X2, p , Xp. Control qr codes data in .netto receive qr code and qr code 2d barcode data, size, image with .net barcode sdkNow, E(Y) can be found from the joint probability distribution of X1, X2, p , Xp as follows. Assume X1, X2, p , Xp are continuous random variables. An analogous calculation can be used for discrete random variables. E1Y2 1c1x1 c2 x2 p cp xp 2 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxpControl qr code image in visual basic.netuse visual .net denso qr bar code printer toconnect qrcode on visual basicx1 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp x2 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp xp fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxpIncoporate linear 1d barcode in .netusing barcode encoding for visual studio .net crystal control to generate, create 1d barcode image in visual studio .net crystal applications.By using Equation 5-24 for each of the terms in this expression, we obtain the following. Bar Code printing with .netuse visual .net crystal bar code implement tobuild barcode for .netMean of a Linear Combination PDF417 writer on .netusing barcode printer for visual .net crystal control to generate, create pdf417 image in visual .net crystal applications.If Y Code 39 drawer with .netusing vs .net crystal todeploy 3 of 9 barcode on asp.net web,windows applicationc1 X1 Visual .net postal alpha numeric encoding technique printer on .netusing barcode creation for visual .net control to generate, create postal alpha numeric encoding technique image in visual .net applications.c2 X2 E1Y2 Control pdf 417 size for .net c# pdf 417 size with visual c#.netcp Xp, c2E 1X2 2 p cp E 1Xp 2 (5-37)Control barcode 39 size with excel spreadsheetsto integrate 3 of 9 barcode and 39 barcode data, size, image with microsoft excel barcode sdkc1E 1X1 2 Java barcode recognizer for javaUsing Barcode reader for Java Control to read, scan read, scan image in Java applications.Furthermore, it is left as an exercise to show the following. Control universal product code version a size in .netto deploy gs1 - 12 and upc a data, size, image with .net barcode sdkCHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS 2d Data Matrix Barcode implement with c#.netusing barcode integrated for visual studio .net (winforms) crystal control to generate, create gs1 datamatrix barcode image in visual studio .net (winforms) crystal applications.Variance of a Linear Combination recognizing barcode in .netUsing Barcode decoder for .net framework Control to read, scan read, scan image in .net framework applications.If X1, X2, p , Xp are random variables, and Y general V1Y 2 c2V1X1 2 1 c2V1X2 2 2 p c2V1Xp 2 p Word Documents ansi/aim code 128 maker for word documentsuse microsoft word code128b integrating todevelop code-128 for microsoft wordc1 X1 2d Matrix Barcode integrated with javausing java toget 2d matrix barcode for asp.net web,windows applicationc2 X2 cp Xp, then in (5-38)2 a a cicj cov1Xi, Xj 2 If X1, X2, p , Xp are independent, V1Y 2 c2V1X1 2 1 c2V1X2 2 2 p c2V1Xp 2 p (5-39)Note that the result for the variance in Equation 5-39 requires the random variables to be independent. To see why the independence is important, consider the following simple examX1. Clearly, X1 and X2 are not indeple. Let X1 denote any random variable and de ne X2 pendent. In fact, XY 1. Now, Y X1 X2 is 0 with probability 1. Therefore, V(Y) 0, regardless of the variances of X1 and X2. EXAMPLE 5-35 In 3, we found that if Y is a negative binomial random variable with parameters p and r, Y X1 X2 p Xr, where each Xi is a geometric random variable with parameter p and they are independent. Therefore, E1Xi 2 1 p and E1Xi 2 11 p2 p2 . From Equation 2 5-37, E1Y 2 r p and from Equation 5-39, V 1Y 2 r11 p2 p . An approach similar to the one applied in the above example can be used to verify the formulas for the mean and variance of an Erlang random variable in 4. EXAMPLE 5-36 Suppose the random variables X1 and X2 denote the length and width, respectively, of a manufactured part. Assume E(X1) 2 centimeters with standard deviation 0.1 centimeter and E(X2) 5 centimeters with standard deviation 0.2 centimeter. Also, assume that the covariance between X1 and X2 is 0.005. Then, Y 2X1 2X2 is a random variable that represents the perimeter of the part. From Equation 5-36, E1Y 2 and from Equation 5-38 V1Y2 22 10.12 2 22 10.22 2 0.04 0.16 0.04 2 2 21 0.0052 0.16 centimeters squared 0.4 centimeters. 2122 2152 14 centimetersTherefore, the standard deviation of Y is 0.161 2 The particular linear combination that represents the average of p random variables, with identical means and variances, is used quite often in the subsequent chapters. We highlight the results for this special case.