If X and Y have a bivariate normal distribution with in .NET Maker qrcode in .NET If X and Y have a bivariate normal distribution with If X and Y have a bivariate normal distribution withDisplay qr codes with .netusing .net vs 2010 touse qr code for asp.net web,windows application0, X and Y are independent. (5-35)Qr Codes scanner on .netUsing Barcode decoder for .net framework Control to read, scan read, scan image in .net framework applications.CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS .net Vs 2010 barcode creator with .netusing barcode implementation for .net vs 2010 control to generate, create barcode image in .net vs 2010 applications.An important use of the bivariate normal distribution is to calculate probabilities involving two correlated normal random variables. EXAMPLE 5-34 Suppose that the X and Y dimensions of an injection-molded part have a bivariate normal distribution with X 0.04, Y 0.08. X 3.00. Y 7.70, and 0.8. Then, the probability that a part satis es both speci cations is P12.95 X 3.05, 7.60 Y 7.802scanning bar code for .netUsing Barcode reader for visual .net Control to read, scan read, scan image in visual .net applications.This probability can be obtained by integrating fXY (x, y; X, Y, X Y, ) over the region 2.95 x 3.05 and 7.60 y 7.80, as shown in Fig. 5-7. Unfortunately, there is often no closed-form solution to probabilities involving bivariate normal distributions. In this case, the integration must be done numerically. EXERCISES FOR SECTION 5-6Qr Bidimensional Barcode printing on visual c#.netusing .net framework toprint qr codes for asp.net web,windows application5-79. Let X and Y represent concentration and viscosity of a chemical product. Suppose X and Y have a bivariate normal distribution with X 4, Y 1, X 2, and Y 1. Draw a rough contour plot of the joint probability density function for each of the following values for : (a) 0 (b) 0.8 (c) 0.8 5-80. Let X and Y represent two dimensions of an injection molded part. Suppose X and Y have a bivariate normal distribution with 0.04, 0.08, 3.00, X Y X 7.70, and Y 0. Determine P(2.95 X 3.05, Y 7.60 Y 7.80). 5-81. In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if speci cations regarding the final color and intensity of light of the lamp are to be met. Let X and Y denote the thickness of two different layers of ink. It is known that X is normally distributed with a mean of 0.1 millimeter and a standard deviation of 0.00031 millimeter, and Y is also normally distributed with a mean of 0.23 millimeter and a standard deviation of 0.00017 millimeter. The value of for these variables is equal to zero. Specifications call for a lamp to have a thickness of the ink corresponding to X in the range of 0.099535 to 0.100465 millimeters and Y in the range of 0.22966 to 0.23034 millimeters. What is the probability that a randomly selected lamp will conform to speci cations 5-82. Suppose that X and Y have a bivariate normal distribution with joint probability density function fXY (x, y; X, Y, X, Y, ). (a) Show that the conditional distribution of Y, given that X x is normal. (b) Determine E1Y 0 X x2 . (c) Determine V1Y 0 X x2 . 5-83. If X and Y have a bivariate normal distribution with 0, show that X and Y are independent. 5-84. Show that the probability density function fXY (x, y; X, Y, X, Y, ) of a bivariate normal distribution integrates to one. [Hint: Complete the square in the exponent and use the fact that the integral of a normal probability density function for a single variable is 1.] 5-85. If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; X, Y, X, Y, ), show that the marginal probability distribution of X is normal with mean X and standard deviation X. [Hint: Complete the square in the exponent and use the fact that the integral of a normal probability density function for a single variable is 1.] 5-86. If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; X, Y, X, Y, ), show that the correlation between X and Y is . 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