h(x) is a decreasing function of x, a similar argument holds. in .NET Develop Denso QR Bar Code in .NET h(x) is a decreasing function of x, a similar argument holds. h(x) is a decreasing function of x, a similar argument holds.QR Code drawer on .netgenerate, create qr code 2d barcode none on .net projectsEXAMPLE S5-3 Let X be a continuous random variable with probability distribution fX 1x2 x , 8 0 x 4 .net Framework qr code recognizer with .netUsing Barcode scanner for .net vs 2010 Control to read, scan read, scan image in .net vs 2010 applications.Find the probability distribution of Y h(X ) 2X 4. Note that y h(x) 2x 4 is an increasing function of x. The inverse solution is x u( y) (y 4) 2, and from this we nd the Jacobian to be J u 1 y2 dx dy 1 2. Therefore, from S5-3 the probability distribution of Y is fY 1 y2 1y 42 2 1 a b 2 y 32 4 , 4 y 12Bar Code generator for .netuse .net framework barcode integrating tobuild barcode on .netWe now consider the case where X1 and X2 are continuous random variables and we wish to nd the joint probability distribution of Y1 h1(X1, X2) and Y2 h2(X1, X2) where the transformation is one to one. The application of this will typically be in nding the probability distribution of Y1 h1(X1, X2), analogous to the discrete case discussed above. We will need the following result.Bar Code barcode library for .netUsing Barcode reader for VS .NET Control to read, scan read, scan image in VS .NET applications.Suppose that X1 and X2 are continuous random variables with joint probability distribution fX1 X2 1x1, x2 2, and let Y1 h1(X1, X2) and Y2 h2(X1, X2) de ne a one-to-one transformation between the points (x1, x2) and (y1, y2). Let the equations y1 h1(x1, x2) and y2 h2(x1, x2) be uniquely solved for x1 and x2 in terms of y1 and y2 as x1 u1(y1, y2) and x2 u2(y1, y2). Then the joint probability of Y1 and Y2 is fY1Y2 1 y1, y2 2 fX1X2 3u1 1 y1, y2 2, u2 1 y1, y2 2 4 0 J 0 (S5-4)Control qrcode image on visual c#use visual studio .net qr bidimensional barcode integrated tocreate quick response code in .net c#where J is the Jacobian and is given by the following determinant: J ` x1 y1, x1 y2 ` x2 y1, x2 y2 Control qr barcode size for .netqr code size on .netand the absolute value of the determinant is used. Control qr bidimensional barcode data in visual basic.net qr code data with vb.netThis result can be used to nd fY1Y2 1 y1, y2 2, the joint probability distribution of Y1 and Y2. Then the probability distribution of Y1 is fY1 1 y1 2 fY1Y2 1 y1, y2 2 dy2Ean13+5 barcode library in .netuse vs .net crystal gtin - 13 printing toconnect ean-13 for .netThat is, fY1 ( y1) is the marginal probability distribution of Y1. EXAMPLE S5-4 Suppose that X1 and X2 are independent exponential random variables with fX1 1x1 2 and fX2 1x2 2 2e 2x2. Find the probability distribution of Y X1 X2 . 2eDisplay 3 of 9 barcode for .netusing barcode maker for vs .net control to generate, create code 39 image in vs .net applications.The joint probability distribution of X1 and X2 is fX1X2 1x1, x2 2 4e Bar Code implement in .netgenerate, create barcode none on .net projects21x1 x22 Insert ucc - 12 for .netusing vs .net todevelop ean/ucc 128 in asp.net web,windows applicationbecause X1 and X2 are independent. Let Y1 h1(X1, X2) X1 X2. The inverse solutions of y1 x1 x2 and y2 x1 x2 y2 11 y1 2 , and it follows that x1 y1 x2 y1 Therefore y2 11 y1 2 2 y2 11 y1 2 2 11 11 y1 1 y1 2 y1 2 y2 c y2 c 11 11 1 y1 2 2 d, x1 y2 x2 y2 c c 11 11VS .NET Crystal international standard serial number writer on .netusing .net crystal torender international standard serial number in asp.net web,windows applicationX1 X2 and Y2 h2(X1, X2) x2 are x1 y1 y2 11 y1 2 and y1 1 d d Control data matrix barcodes image with .netusing windows forms toconnect datamatrix 2d barcode in asp.net web,windows applicationy1 2 y1 2 Control barcode 3/9 image on .netgenerate, create 3 of 9 none on .net projects1 d, y1 2 2 Control barcode 3/9 data for javato incoporate code 3/9 and uss code 39 data, size, image with java barcode sdky2 y1 2 2 Include qr-code in visual c#.netuse visual .net quick response code printer toadd qr-codes with visual c#and from Equation S5-4 the joint probability distribution of Y1 and Y2 is fY1Y2 1 y1, y2 2 fX1X2 3u1 1 y1, y2 2, u2 1 y1, y2 2 4 0 J 0 y2 4e 23 y1 y2 11 y12 y2 11 y124 ` ` 11 y1 2 2 4eControl ean / ucc - 14 image with visual c#using .net toaccess ean 128 in asp.net web,windows applicationy2 11Control pdf417 image in visual basic.netgenerate, create barcode pdf417 none for visual basic.net projectsy1 2 2Ean 128 Barcode scanner in noneUsing Barcode Control SDK for None Control to generate, create, read, scan barcode image in None applications.for y1 0, y2 0. We need to nd the distribution of Y1 ability distribution of Y1, or fY1 1 y1 2 fY1Y2 1y1, y2 2 dy2 4eBarcode barcode library for exceluse excel barcode development toget bar code on excelX1 X2 . This is the marginal prob-3y2 11 y1 y1 2 2 4 dy2 0 y1 2 2An important application of Equation S5-4 is to obtain the distribution of the sum of two independent random variables X1 and X2. Let Y1 X1 X2 and let Y2 X2. The inverse solutions are x1 y1 y2 and x2 y2. Therefore, x1 y1 x2 y1 1 0 x1 y2 x2 y2 1 1and J 1. From Equation S5-4, the joint probability density function of Y1 and Y2 is fY1Y2 1 y1, y2 2 f X1 1 y1 y2 2 f X2 1 y2 2Therefore, the marginal probability density function of Y1 is fY1 1 y1 2 fX1 1 y1 y2 2 fX2 1 y2 2 dy2The notation is simpler if the variable of integration y2 is replaced with x and y1 is replaced with y. Then the following result is obtained.