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University of Colorado Department of Civil, Environmental and Architectural Engineering Statistical Methods for Water and Environmental Engineers CVEN-5454 Spring 2007 Midterm Make up Question Date: 05/05/2007 20 points __________________________________________________________________________________________________ Please write the steps clearly so that points can be awarded even when the numerical answer is incorrect. 1. A random variable R has the following density function f(r) = k * (r – 10)*(20 - r) for 10 <= r <=20 Determine: (a) the value of k (b) Median of R [3,4] A system of storm sewers is proposed for a city. The probability mass function for the number of occurrences of rainstorm each year in the city is #of rain storms in a year 0 1 2 3 Probability 0.3 0.4 0.2 0.1 (c) What is the mean number of rainstorms in a year for the city? The maximum runoff rate in each storm is lognormal with a median of 7 cfs and coefficient of variation of 15%. The proposed sewer system is shown to be adequate for any storm with runoff rate less than 8 cfs. Assume that the maximum runoff rates between storms are statistically independent. (d) What is the probability of flooding during a rainstorm? [2, 5] From 18 years of record in a city, the number of type I and type II tornadoes are 9 and 54, respectively. The occurrences of each type of tornado are assumed to be statistically independent and follow a Poisson distribution. (e) what is the probability that there will be exactly 2 tornadoes in the city next year? (f) What is the probability of at least 1 tornado of type I in the next 5 years [4,2]