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Unformatted text preview: Florida International University Department of Civil and Environmental Engineering CWR6117 – Statistical Hydrology Assignment No. 1 – Due February 3rd, 2009 The problems in this assignment are intended to give you an idea of the basic concepts you will need for this class. Solutions to many of these problems can be found in basic statistics texts, or even perhaps on the web. However, I strongly recommend that you try to solve them with as little assistance as possible so you can discover what you need to learn and review. If you are unsure of the proper approach, the best advice is to go back to fundamental principles and rederive the appropriate theory. Problem 1: A gambler asks you to join a dice game. He will give you $10 if you can roll 2 sevens in three successive tries. You will give him $5 if you fail. Should you play? What is your expected return if you do? Problem 2: Calculate the mean and variance of the following probability density function (pdf): f X ( x) 2 x 0 x 1 f X ( x) 0 otherwise Problem 3: The solution to a steady‐state groundwater flow equation with constant hydraulic conductivity and random boundary conditions is given by: x
h( x) a (b a) L
where L is the length of the solution domain, x is the distance from the boundary located at the origin, a is the boundary head at x=0, b is the boundary head at x=L. What are the mean and variance of the head distribution h(x) if the mean of a is 1.0, the mean of b is 2.0 and the standard deviation of both a and b is 0.5? The velocity of groundwater can be calculated using Darcy’s equation: K dh
v n dx
What are the mean and variance of the groundwater velocity is the ratio of hydraulic conductivity K to porosity n is equal to 0.1? Problem 4: The probability that a Miami‐area college student will get a job if she/he graduates from FIU is 0.9. The probability that the student will get a job if she/he did not graduate from FIU is 0.3. The probability that she/he will graduate from FIU is 0.1. What is the probability that a Miami‐area college student will get a job? Problem 5: The probability of correctly detecting a contaminant in a water sample is 0.80. The probability of a “false alarm” (detection when no contaminant is present) is 0.10. You are presented with a group of 10 samples. You know that exactly 3 of these have been contaminated. You select one of the 10 samples at random and detect contamination. What is the probability that the sample is actually contaminated. Problem 6: If the probability density function (pdf) of variable X is: f X ( x) e x and Y=aX+b; what is the pdf of Y? Problem 7: Derive the distribution of Z=X+Y if X is uniformly distributed over the interval 0<x<2 and Y is uniformly distributed over the interval 1<Y<3. Problem 8: Suppose that a random variable X is known to have the following exponential distribution: f X ( x) ae ax 0 x Derive the distribution of X if a is a random variable uniformly distributed in the interval 0<a<1. Problem 9: Suppose that X, Y and Z are three independent, identically distributed streamflows (observed during different storm events) with a common probability density function (pdf) given by: f X ( x ) e x where is a specified parameter. What is the pdf of the maximum streamflow W? Graph the pdfs for both X and W on the same axes. What does this plot suggest about the probability of a rare flood event? Hint: note that if W is the maximum of (X,Y,Z) then X, Y and Z must be less than or equal to W. Problem 10: Suppose that a random process Xi is described by the difference equation: X i X i 1 i where X0=0, and i is a zero‐mean, unit variance random variable that is uncorrelated with j for i≠j. What are the mean and variance of Xi? What is the correlation between Xi and Xi1 as i approaches infinity? ...
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This note was uploaded on 07/09/2011 for the course CWR 6117 taught by Professor Miralles during the Spring '09 term at FIU.
- Spring '09