University of Colorado
Department of Civil, Environmental and Architectural Engineering
Statistical Methods for Water and Environmental Engineers
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
(a) the value of k
(b) Median of R
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
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
(f) What is the probability of at least 1 tornado of type I in the next 5 years