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Unformatted text preview: UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil and Environmental Engineering Department CEE 110 Introduction to Probability and Statistics for Engineers Spring Quarter 2011 MW 122 PM Franz 1178 Prof. K. D. Stolzenbach 5732J Boelter Hall, 2067624 [email protected] Problem Set 3 (due Wednesday April 20, 2011) _____________________________________________________________________________ 1. (problems 3.16, 3.34, and 3.50 in text) A discrete random variable X has a probability mass function: f ( x ) = (8/7)(1/2) x , x = 1, 2, 3 a) Determine the following probabilities (i) P ( X ≤ 1) (ii) P ( X > 1) (iii) P (2 < X < 6) (iv) P ( X ≤ 1 or X > 1) [4/7, 3/7, 1/7, 1] b) Determine the cumulative distribution function and the following probabilities i P ( X < 1.5) ii P ( X ≤ 3) iii P ( X > 2) iv P (1 < X ≤ 2) [4/7, 1, 1/7, 2/7] c) Determine the mean and variance of the random variable x . [11/7, 0.53] 2. ( based on problems 4.4 and 4.18 in the text) The diameter X of a particle of contamination (in micrometers) is modeled with the continuouse probability function f ( x ) = 3/ x 4 for x > 1. a) Determine the following probabilities i P ( X < 2) ii P ( X > 5) iii P (4 < X < 8) iv P (X< 4 or X > 8) [0.875, 0.008, 0.014, 0.986] b) Determine the cumulative distribution function F( x ) c) Determine the mean and variance of the random variable x . [3/2, 3/4] 3. Assume that X and Y are independent measurements with uncertainties Δ X = 0.2 and Δ Y = 0.4. Find the uncertainties in the following quantities: a) 3 X b) X – Y c) 2 X + 3 Y [0.6, 0.45, 1.3] 4. One way to measure the water content of a soil is to weigh the soil both before and after drying it in an oven. The water content is W = ( M 1 – M 2 )/ M 1 , where M 1 is the mass before drying and M 2 is the mass after drying. Assume M 1 = 1.32 ± 0.01 kg and M 2 = 1.04 ± 0.01 kg. a) Estimate W and the uncertainty in the estimate Δ W b) Which would provide a greater reduction in the uncertainty in W : reducing the uncertainty in M 1 to 0.005 kg or reducing the uncertainty in M 2 to 0.005 kg?...
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 Spring '11
 Yukov
 Normal Distribution, Probability, Environmental Engineering, Probability theory

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