Convolution of X1 and X2 in .NET Attach QR Code 2d barcode in .NET Convolution of X1 and X2 Convolution of X1 and X2Add qr-codes in .netusing .net framework tocompose qrcode on asp.net web,windows applicationIf X1 and X2 are independent random variables with probability density functions fX1(x1) and fX2 (x2), respectively, the probability density function of Y X1 X2 is fY 1 y2 fX1 1 y x2 f X2 1x2 dx (S5-5)QR Code decoder on .netUsing Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.The probability density function of Y in Equation S5-5 is referred to as the convolution of the probability density functions for X1 and X2. This concept is commonly used for transformations (such as Fourier transformations) in mathematics. This integral may be evaluated numerically to obtain the probability density function of Y, even for complex probability density functions for X1 and X2. A similar result can be obtained for discrete random variables with the integral replaced with a sum. In some problems involving transformations, we need to nd the probability distribution of the random variable Y h(X) when X is a continuous random variable, but the transformation is not one to one. The following result is helpful.reading barcode with .netUsing Barcode scanner for .net vs 2010 Control to read, scan read, scan image in .net vs 2010 applications.Suppose that X is a continuous random variable with probability distribution fX (x), and Y h(X) is a transformation that is not one to one. If the interval over which X is de ned can be partitioned into m mutually exclusive disjoint sets such that each of the inverse functions x1 u1( y), x2 u2( y), p , xm um( y) of y u(x) is one to one, the probability distribution of Y is fY 1 y2 where Ji u 1 y2 , i i a fX 3ui 1 y2 4 0 Ji 0Bar Code barcode library in .netuse visual .net bar code encoding toinclude bar code for .net(S5-6)Control qr code data for visual c#.net qr code jis x 0510 data for visual c#1, 2, p , m and the absolute values are used. Qr-codes barcode library in .netgenerate, create qrcode none with .net projectsTo illustrate how this equation is used, suppose that X is a normal random variable with mean and variance 2, and we wish to show that the distribution of Y 1X 2 2 2 is aControl qr code 2d barcode size in vb.net qr-codes size on visual basicchi-squared distribution with one degree of freedom. Let Z probability distribution of Z is the standard normal; that is, f 1z2 1 e 12GS1-128 barcode library on .netusing visual .net crystal toencode ucc-128 with asp.net web,windows applicationz2 2Make code 3/9 for .netgenerate, create bar code 39 none for .net projects, and Y Qrcode barcode library in .netuse .net framework crystal qr code jis x 0510 encoding todraw qr with .netZ 2. The EAN13 barcode library on .netgenerate, create european article number 13 none for .net projectsThe inverse solutions of y z2 are z 1y, so the transformation is not one to one. De ne z1 1y and z2 1y so that J1 11 22 1y and J2 11 22 1y. Then by Equation S5-6, the probability distribution of Y is , z fY 1 y2 1 1 1 1 e y2` ` e y2` ` 12 21y 12 21y 1 y1 2 1 e y 2, y 0 12 2 1 11 22 , so we may write f( y) as 1 2Bookland EAN implement for .netgenerate, create isbn bookland ean none with .net projectsNow it can be shown that 1 fY 1 y2 Control gs1 datamatrix barcode size on java barcode data matrix size in java1 a b 2 SQL Server qrcode printer on .netusing sql 2008 toproduce qr code in asp.net web,windows applicationy1 2Control ucc - 12 data in office excelto incoporate upc-a supplement 5 and upc symbol data, size, image with excel spreadsheets barcode sdkwhich is the chi-squared distribution with 1 degree of freedom. EXERCISES FOR SECTION 5-8Control ecc200 image on wordusing word documents tomake data matrix ecc200 for asp.net web,windows applicationS5-1. Suppose that X is a random variable with probability distribution fX 1x2 1 4, x 1, 2, 3, 4 S5-5. A current of I amperes ows through a resistance of R ohms according to the probability distribution fI 1i2 2i, 0 i 1Control code 39 image on .netusing barcode generating for winforms control to generate, create uss code 39 image in winforms applications.Find the probability distribution of the random Y 2X 1. S5-2. Let X be a binomial random variable with p 0.25 and n 3. Find the probability distribution of the random variable Y X 2. S5-3. Suppose that X is a continuous random variable with probability distribution fX 1x2 x , 18 0 x 6Barcode integrating in .netuse rdlc report files barcode implement toprint barcode with .netSuppose that the resistance is also a random variable with probability distribution fR 1r2 1, 0 r 1 Control gs1 barcode size on visual c#to display ean/ucc 128 and data, size, image with visual c#.net barcode sdk(a) Find the probability distribution of the random variable Y 2X 10. (b) Find the expected value of Y. S5-4. Suppose that X has a uniform probability distribution fX 1x2 1, 0 x 1UCC-128 creation for visual basic.netgenerate, create gs1 128 none for vb.net projectsAssume that I and R are independent. (a) Find the probability distribution for the power (in watts) P I 2R. (b) Find E(P). S5-6. A random variable X has the following probability distribution: fX 1x2 e x, x 0 X 2. X1 2. ln X.(a) Find the probability distribution for Y (b) Find the probability distribution for Y (c) Find the probability distribution for YShow that the probability distribution of the random variable Y 2 ln X is chi-squared with two degrees of freedom.S5-7. The velocity of a particle in a gas is a random variable V with probability distribution fV 1v2 av2e