Question

The following emprical equation is derived for the solution of an engineering problem: Z = X Y^2 √ W where:     X : Uniformly distributed between 2.0 and 4.0, Y: Normally distributed with median 1.0 and Pr(Y ≤ 2.0) = 0.9207, W: Exponentially distributed with a median of 1.0, and X, Y and W are statistically independent.

a) Compute the mean values, variances and coefficients of variation of X, Y and W, respectively.

b) Compute the mean, standard deviation and coefficient of variation of Z using the first-order approximation.  

Answer 1

X = unif (2,4)

mean = (a+b)/2

variance = (b-a)^2/12

sd =sqrt(variance)

coefficient of variation = sd/ mean

Y =normal with median = 0 and P(Y < 2 )= 0.9207

mean = median for normal distribution = 1

P(Z< z ) = 0.9207

z = 1.4098

(2 - 1)/sigma = 1.4098

sigma = 1/1.4098 = 0.7093

variance = sigma^2 = 0.7093^2 = 0.5031

coefficient of variation =sd/mean = 0.7093 /1 = 0.7093

Z = median = 1

ln 2 / lambda = 1

hence 1/lambda = 1.4427

hence mean = 1/lambda = 1.4427

sd = 1/lambda = 1.4427

variance = 1.4427^2 = 2.0814

coefficient of variation = sd/mean = 1