If X and Y have a bivariate normal distribution with in .NET

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If X and Y have a bivariate normal distribution with
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0, X and Y are independent. (5-35)
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CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS
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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 7.802
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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 5-6
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5-79. 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 . [Hint: Complete the square in the exponent].
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