Suppose terrorists dropped C units of a toxic chemical (of negligible

CIVL 181 MIDTERM EXAMINATION 2 April 21, 2008

Please answer all 3 problems on the blank space provided in this booklet

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Problem 1 (24%) Circle T (true) or F (false) for each of the following statement. A correct answer will score 3 points; whereas 1 point will be deducted for each incorrect answer. T

F If the occurrence of polluted days follows a Bernoulli Sequence model; the mean return period of a polluted day may be 5.2 days.

T

F The sum of 10 lognormal random variables will follow a normal distribution.

T

F The sum of 5 independent exponential random variables, each with the same parameters, is more likely to follow a gamma than a normal distribution.

T

F Suppose W=X+Y, both X and Y are normally distributed, and the variance of W is 10. If an engineer discovered that Y is actually log normally distributed but with the same variance as before, the variance of W will be still 10.

T

F An engineer evaluates a problem involving two correlated discrete random variables X and Y. Suppose he needs to calculate the probability of Y>2, but he already knew that X is equally likely to be 5 or 6. He would need to use the conditional distribution of Y given X=5.5

T

F In general, one can determine the joint PMF of X and Y, provided that both the marginal PMF’s of X and Y are given.

T

F The sum of a discrete and a continuous random variable is a continuous random variable.

T

F If X and Y are correlated, its correlation coefficient can be -0.5.

Problem 2 (40%) Before the Olympic Games open this summer, the Olympic torch will pass through Hong Kong. Suppose the distance that it will go through is 50km. The normal time taken for this trip may be described by a normal random variable with a mean of 4 hours and a standard deviation of 1 hour. However disturbances can occur during the trip at a mean rate of one per 50 km. Each incident of disturbance will cause a delay for the trip. The delay may be described by a normal random variable, with both mean and standard deviation equal to 0.5 hour. Suppose the occurrence of disturbances follows a Poisson process, and the delay times among incidences of disturbance as well as the normal time are all statistically independent. (a) If indeed only one disturbance will occur in the whole trip, what is the probability that the trip can still be completed within 4 hours? (b) Repeat part (a) if the number of disturbances is not known, but simply follows the Poisson process as described previously. (c) Now assume that 10 disturbances will take place, and the delay time caused by each disturbance follows an exponential distribution. What is the probability that the torch event will go beyond midnight. Assume the event begins at 10 in the morning? State clearly if there is any assumption that you need to make and justify accordingly. Problem 3 (36%) Consider the total weight of audiences in a concert hall. The mean and c.o.v. of a male audience’s weight are 60 kilos and 25% respectively; whereas those for a female audience are 50 and 10% respectively. Suppose the hall is fully occupied with 100 audiences. The percentage of female is uncertain, but has a mean of 70% and a c.o.v. of 20%. (a) Write down the formula to calculate the total weight W of people in the hall. Define the random variables involved. Assume all the boys weigh the same, and all the girls weigh the same in this part. (b) Determine the mean, variance and c.o.v. of W. Are these values exact or approximate? (c) If the weights of each boy and each girl are all statistically independent, will you expect any changes on the mean and variances obtained in Part b? Please explain.