a. Explanation Calculation: It is given that the water retention curves for a range of soil textures follows a lognormal distribution with the parameters θ = − 3.8 and ω = 0.7 . z-score: If a random variable X follows large sample distribution with mean μ and σ , the z -score is defined as, Z = X − μ σ follows standard normal distribution . Lognormal distribution: If W follows normal distribution with mean θ and the variance ω 2 , then X = exp ( W ) follows lognormal distribution with probability density function. f ( x ) = 1 x ω 2 π e x p [ − ( ln ( x ) − θ ) 2 2 ω 2 ] , 0 < x < ∞ with mean E ( X ) = exp ( θ + ω 2 2 ) and variance V ( X ) = exp ( 2 θ + ω 2 ) ( exp ( ω 2 ) − 1 ) . The value of P ( X < 0.02 ) : P ( X < 0.02 ) = P ( exp ( W ) < 0.02 ) = P ( W < ln ( 0 b. To determine Find the value of x for which P ( X ≤ x ) = 0.95 . c. To determine Find the mean and the variance of X .

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

See similar textbooks

Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

Videos

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 4.11, Problem 180E

To determine

Find the value of P(X<0.02) .

To determine

Find the value of x for which P(X≤x)=0.95.

To determine

Find the mean and the variance of X.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.11, Problem 179E

Chapter 4.11, Problem 181E

Students have asked these similar questions

Could you please answer this question using excel.Thanks

Questions An insurance company's cumulative incurred claims for the last 5 accident years are given in the following table: Development Year Accident Year 0 2018 1 2 3 4 245 267 274 289 292 2019 255 276 288 294 2020 265 283 292 2021 263 278 2022 271 It can be assumed that claims are fully run off after 4 years. The premiums received for each year are: Accident Year Premium 2018 306 2019 312 2020 318 2021 326 2022 330 You do not need to make any allowance for inflation. 1. (a) Calculate the reserve at the end of 2022 using the basic chain ladder method. (b) Calculate the reserve at the end of 2022 using the Bornhuetter-Ferguson method. 2. Comment on the differences in the reserves produced by the methods in Part 1.

Calculate the correlation coefficient r, letting Row 1 represent the x-values and Row 2 the y-values. Then calculate it again, letting Row 2 represent the x-values and Row 1 the y-values. What effect does switching the variables have on r? Row 1 Row 2 13 149 25 36 41 60 62 78 S 205 122 195 173 133 197 24 Calculate the correlation coefficient r, letting Row 1 represent the x-values and Row 2 the y-values. r=0.164 (Round to three decimal places as needed.) S 24

Chapter 4 Solutions

Applied Statistics and Probability for Engineers

Ch. 4.2 - 4-1. Suppose that f(x) = e−x for 0 < x. Determine...Ch. 4.2 - 4-2. Suppose that f (x) = 3(8x – x2)/256 for 0< x...Ch. 4.2 - 4-3. Suppose that f (x) = 0.5 cos x for −π/2 < x <...Ch. 4.2 - Prob. 4E Ch. 4.2 - 4-5. Go Tutorial Suppose that for 3 < x < 5....Ch. 4.2 - Prob. 6E Ch. 4.2 - 4-7. Suppose that f(x) = 1 .5x2 for −1 < x < 1....Ch. 4.2 - 4-8. The probability density function of the time...Ch. 4.2 - 4-9. The probability density function of the net...Ch. 4.2 - Prob. 10E

Knowledge Booster

$Statistics$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Find the value of P ( X &lt; 0.02 ) .

Concept explainers

Videos

Find the value of P(X<0.02) .

Find the value of x for which P(X≤x)=0.95.

Find the mean and the variance of X.

Want to see the full answer?

Chapter 4 Solutions

Find the value of P ( X < 0.02 ) .