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p196_exercices_5-6

# p196_exercices_5-6 - 196 cums JOINT PROBABILITY orsrmsunons...

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Unformatted text preview: 196 cums JOINT PROBABILITY orsrmsunons 5-79. Determine the value of c such that the function f(x,y)=cx2y for0<x<3 and 0<y<2 satisﬁes the properties of a joint probability density function. Determine the following: (a) P(X< 1, r< 1) (b) P(X< 2.5) (c)P(1 <r< 2.5) (d) P(X> 2,1< r< 1.5) (a) E(X) (f) 30’) (g) Marginal probability distribution of the random variableX. (h) Conditional probability distribution of Y given that)! = 1. (i) Conditional probability distribution of X given that Y = 1. 5-80. The joint distribution of the continuous random vari- ables X, Y, and Z is constant over the region 12 + y2 5 1, 0 <2 < 4. Determinethe following: (a) P(X1+ +r2 < 0.5) (b) P(X2 + risoqs z< 2) (c) Joint conditional probability density ﬁmction of X and 1’ given that Z = (d) Marginal probabiiity density ﬁmction of X? (e) Conditional mean of 2 given thatX = 0 and Y = 0. (f) Conditional meanongiventhatX= rand Y =y. 5-81. Suppose that Xand Yare independent, continuous lmiformrandomvariablesforo <x < lme <y<l. Use the joint probability density function to determine the proba— bility that [X— Y] < o 5. 5-82.1he1ifetimesofsixmajorcomponentsinacopiqate independentexponentialrandomvariableswiﬂlmeansofSOOO, 10,000,10,,000 20,000, 20,,000 and25,000hours,respectively. (a) Whatistheprobahility thatthelifetimes of allthecornpo- nentsexceedSOOOhours‘? (b) What is the probability that at least one component life- time exceeds 25,000 hours? 5-83. Contamination problems in semiconductor manufac- turing can result in a ﬁmctional defect, a minor defect, or no defectinthe ﬁnalproduct. Suppose that20, 50, and 30% ofthe contamination problems result in flmctional, minor, and no de- fects, remectively. Assume that the defects of 10 contamina- tion problems are independent. (a) Whatislhepmbabilitythettbe 10 contaminationproblems result in two ﬁmctional deﬁcts and ﬁve minor defects? (b) What is the distribution of the number of contamination problems that result in no defects? (c) What is the expected nmnber of contamination problems that result in no defects? 5-34. The weight of adobe bricks for construction is normaﬂydistributedndﬂiamean of3 pounds andastandard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen. (a) What is the probability that the mean weight of the sample is less than 2.95 pounds? (b) What value will the mean weight exceed with probability 0.99? 5-85. Thelengthandwidthofpanelsused forinterior ... (in inches)aredenowdasXand Y, respectively. Suppose I. .7 and Y are independent, continuous uniform random variables- 17.75 <x<1825and475<y<525rwpectively (a) By integrating the joint probability density function - the appropriate region, determine the probability that , area of a panel exceeds 90 square inches. (b) What is the probability that the perimeter of s p:i exceeds 46 inches? 5-86. The weight of a small candy is normally distri ' with a mean of 0.1 otmce and a standard deviation of I.: ounce. Suppose that 16 candies are placed in a package .3 that the weights are independent. (a) What are the mean and variance of package net (b) Whatis the probability that the net weight ofa pat: as: less than 1.6 ounces? (c) If 17 candies are placed in each package, what is i probability that the net weight of a package is less m" 1.6 ounces? 5-87. The time for an automated system in a warehouse: locate apartisnormally distributedwithamean of45 s- n and a standard deviation of 30 seconds. Suppose that .5 pendent requests are made for 10 parts. (a) What is the probability that the average time to locate parts exceeds 60 seconds? (b) What is the probability that the total time to locate parts exceeds 600 seconds? 5-88. A mechanical assembly used in an automobile - I 1:- contains four major components. The weights of u' components are independent and normally distributed . ' the following means and standard deviations (in ounces): 4 0.4 Right case 5.5 0.5 Bearing assembly 10 0.2 Bolt assembly 8 0.5 (a) What is the probability that the weight of an assemb exceeds 29.5 ounces? (b) What is the probability that the mean weight of ei_- . independent assemblies exceeds 29 ounces? 1 5-89. SupposeX and Y have a himste normal distributiorf “max—4 (Ty—1p“: 4,|liy= 4,3ﬂdp="ﬂ.2.m a rough contour plot of the joint probability density ﬁlnction. 5'90 IffMXJ) = “1—;-6XD {37—2 [(1 — Ilz — 1.6(x — l)(y — 2) + (y — 2W} ...
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