CE 463 Probabilistic and Statistical Methods in Civil Engineering (S2) August 25, 2016 Topic – Joint Probability Distribution, Functions of Random Variables Practice Problems 1. The absolute velocity (X) of particles in a gas follows the PDF x2 4x2 f X x 3 exp 2 for x 0 a a 0
elsewhere
where a is a constant. Determine the PDF of Y for the particle kinetic energy Y
1 mX 2 , 2
where m is the mass of a particle. 2. Bob and John are travelling from city A to city D. Bob decides to take the upper route (through B), whereas John takes the lower route (through C) as shown below:
The travel times (in hours) between the cities indicated are normally distributed as follows: T1 ~ N(6,2), T2 ~ N(4,1), T3 ~ N(5,3), and T4 ~ N(4,1). Although the travel times can generally be assumed to be statistically independent, T3 and T4 are dependent with a correlation coefficient of 0.8. (a) What is the probability that John will not arrive in city D within 10 hours? (Ans – 0.397) (b) What is the probability that Bob will arrive in city D earlier than John by at least 1 hour? (Ans – 0.327) (c) Which route (upper or lower) should be taken if one wishes to minimize the expected travel time from A to D? Justify (Ans – lower) 3. Consider a long cantilever structure shown in the figure below, which is loaded by a force F at a distance X from the fixed end A. Suppose F and X are independent lognormal random variables with mean values of 0.2 kips and 10 ft., respectively. The coefficient of variations are 20% and 30%, respectively.
(a) Determine the probability that the bending moment at A will exceed 3 kip-ft? (Ans – 0.093) (b) Suppose there are 50 forces, each acting at a random location along the beam. Each force has a lognormal distribution with a mean of 0.2 kip and c.o.v. of 20%, and the location of each force is also lognormal with a mean of 10 ft. from A and a c.o.v. of 30%. Assume statistical independence between individual values of the 50 forces and also between individual locations of the 50 forces. Determine the probability that the overall induced bending moment at A will exceed 120 kip-ft. (Ans – 5.5×10 -5) Page 1 of 2
4. A shuttle bus operates from a shopping center, travels to owns A and B sequentially, and then returns to the shopping center as shown in the figure below.
Assume that the respective travel times (under normal traffic) are independent Gaussian random variables with the following statistics:
However, during rush hours (8:00-10:00 AM and 4:00-6:00 PM), the mean travel time between Town A and Town B will increase by 50% with the same for their coefficient of variation. (a) The scheduled time for one round trip of the shuttle bus is 2 hours. Determine the probability that a round trip will not be completed on schedule under normal traffic. (b) What is the probability that a passenger, originating in Town A, will arrive at the shopping center within 1 hour under normal traffic? (c) The number of passengers originating from Town B is twice that of the number originating from Town A. Calculate the percentage of the passengers originated from both Town A and B will arrive at the shopping center during rush hours in less than 1 hour. (d) A passenger starting from Town B has an appointment at 3:00 PM at the shopping center. If the bus left Town B at 2:00 PM but has not arrived at the shopping center at 2:45 PM, what is the probability that he or she will arrive on time for his/her appointment? 5. A contractor submits bids to three highway jobs and two building jobs. The probability of winning each job is 0.6. The profit from a highway job is normally distributed as N(100, 40) in thousand dollars, whereas that for a building job is N(80, 20) in thousand dollars. Assume that winning each job is an independent event and the profits between jobs are also statistically independent. (a) What is the probability that the contractor will win at least two jobs? (b) What is the probability that he will win exactly one highway job but none of the building jobs? (c) Suppose he has won two highway jobs and one building job. What is the probability that the total profit will exceed 300 thousand dollars? (d) How would the answer in part (c) change if the profit from the two highway jobs are correlated with a correlation coefficient of 0.8?
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