CEE 110 PS3 11

CEE 110 PS3 11 - UNIVERSITY OF CALIFORNIA LOS ANGELES Civil...

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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 12-2 PM Franz 1178 Prof. K. D. Stolzenbach 5732J Boelter Hall, 206-7624 [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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CEE 110 PS3 11 - UNIVERSITY OF CALIFORNIA LOS ANGELES Civil...

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