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Get Full Access to Probability And Statistics For Engineers And The Scientists - 9 Edition - Chapter 4 - Problem 14e
Get Full Access to Probability And Statistics For Engineers And The Scientists - 9 Edition - Chapter 4 - Problem 14e

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# The article “Modeling Sediment and Water Column ISBN: 9780321629111 32

## Solution for problem 14E Chapter 4

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Problem 14E

The article “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants” (?Water Research, ?1984: 1169–1174) suggests the uniform distribution on the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a.? ?What are the mean and variance of depth? b.? hat is the cdf of depth? c.? ?What is the probability that observed depth is at most 10? Between 10 and 15? d.? ?What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations?

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Step 1 of 3

Answer: Step1: A uniform distribution on the interval [A,B] if the pdf of ‘X’ is Given, the uniform distribution on the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. That is, Step2: a). To find the mean and variance of depth. 20 Mean = E(x) = xf(x) dx 7.5 20 1 = x 207.5dx 7.5 20 = x dx 7.5 12.5 20 = 1 x dx 12.57.5 1 x1+1 20 = 12.5 1+1] 7.5 = 1 ( ) 20 12.5 2 7.5 1 2 2 = 25 ((7.5) (20) ) = 13.75. 20 V(x) = =2 (x 13.75) f(x) dx 7.5 20 = (x 13.75) 2 1 dx 207.5 7.5 2 2 2 Here we using the formula (a b) = a 2ab + b 20 = 1 (x 2(13.75) x + (13.75) ) dx2 12.5 7.5 1 x3 20 2 20 2 20 = 12.5([ 3 7.5 27.5[x ] 7.5+ (13.75) [x] 7.5 1 (20) (7.5) 2 2 = 12.5 3 ] 27.5[(20) (7.5) ] + 189.0625[20 7.5] = 1 (162.76) 12.5 = 13. Standard deviation = 13 = 3.6056. Therefore, the mean and variance of depth is E(x) = 13.75 and variance V(x) = 13. Step3: b). The aim is to obtain the cumulative distribution function. The cumulative distribution function is x F(x) = 1 dx 7.5 207.5 = 1 [x] x 12.5 7.5 1 = 12.5[x 7.5] = x7.5 12.5 20 1 F(x) = 207.5x 7.5 = 1 [x] 20 12.5 7.5 1 = 12.5[20 7.5] 12.5 = 12.5 = 1 Then, the cumulative distribution function of depth is given by Step4: c). To find the probability that observed depth is at most 10. That is, 107.5 P(x < 10) = F(10) = 12.5 = 0.2. Therefore, the probability that observed depth is at most 10 is 0.2. Then, To find the probability that observed depth is between 10 and 15. P(10 < x < 15) = F(15) - F(10) 157.5 107.5 = 12.5 - 12.5 = 0.6 - 0.2 = 0.4. Therefore, the probability that observed depth is between 10 and 15 is 0.4. Step5: d). i).To find the probability that the observed depth is within 1 standard deviation. That is, E(x) ± 1 13.75 ± 1(3.6056) (10.1444,17.3565) Therefore, P(- < X< + ) F(17.3565) - F(10.1444) 17.35657.510.14447.5 12.5 - 12.5 0.7885 0.2116 0.5769 Therefore, the probability that the observed depth is within 1 standard deviation is 0.5769 ii). Next to find the probability that the observed depth is within 2 standard deviation . That is, E(x) ± 2 13.75 ± 2(3.6056) (6.5388,20.9612). Therefore, P(-2 < X< +2 ) F(20.9612) - F(6.5388) 20.96127-56.53887.5 12.5 12.5 1.0769- (-0.0768) 1.1537 Therefore, the probability that the observed depth is within 2 standard deviation is 1.1537.

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##### ISBN: 9780321629111

This full solution covers the following key subjects: depth, within, standard, sediment, Water. This expansive textbook survival guide covers 18 chapters, and 1582 solutions. Since the solution to 14E from 4 chapter was answered, more than 2519 students have viewed the full step-by-step answer. Probability and Statistics for Engineers and the Scientists was written by and is associated to the ISBN: 9780321629111. The answer to “The article “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants” (?Water Research, ?1984: 1169–1174) suggests the uniform distribution on the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a.? ?What are the mean and variance of depth? b.? hat is the cdf of depth? c.? ?What is the probability that observed depth is at most 10? Between 10 and 15? d.? ?What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations?” is broken down into a number of easy to follow steps, and 94 words. The full step-by-step solution to problem: 14E from chapter: 4 was answered by , our top Statistics solution expert on 05/06/17, 06:21PM. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9.

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