1 Econ 310 Professor Wallace Fall 2013 Homework #5* 1. River Dam Project. Especially in the western part of the United States, needs from municipal, environmental and agricultural interests place an exceptionally high demand on current stocks of water. Among other negative impacts, reducing the flow of water through a riparian ecosystem at critical times of the year has been shown to alter the ability of birds to nest in areas safe from predators. If water flows during these critical times of the year are too low, the results can often be catastrophic for birds with populations plummeting and species approaching extinction. Whooping Cranes are one endangered species affected by low water flows in the western United States. In the Spring, the Cranes nest along a small stretch of the Platte River in Nebraska, eating toads, snails and leftover corn in the Nebraska fields in preparation for their migration into Artic breeding grounds. Because of the Endangered Species Act and the reliance of the cranes on water flows in the Platte River, cities, farmers and environmentalists often struggle to find solutions that serve the needs of the cranes, while also giving cities and farmers the water that they need at all times of the year. The cranes need high flows in the spring from snow runoff in the Rocky Mountains to clear shrub along the banks of the Platte. Cities need to dam the river both for electricity generation and for the storage of water for consumption. Farmers need to dam the river to help them regulate supplies for their own consumption including crop irrigation at various times of the year. These three groups are often conflicting over their water flow demands from the Platte River. Suppose you are serving on a multi-state regulating board in charge of assessing the water needs of consumers of Platte River water, including municipalities, agricultural entities and ecosystems. The panel includes experts in water flow control, as well as biologists and representatives of all groups interested in water supplies. Your understanding of statistics should give you the basic tools necessary to engage in a meaningful dialogue with the experts. Assume that the annual flow of the Platte River is normally distributed with a mean of 1,900 cubic-feet per second (cfs). a) If the standard deviation (square root of variance) is 800 cfs, what is the probability that the flow exceeds 2,000 cfs? b) A flow duration curve recorded the highest instantaneous flow of 38,500 cfs for the Platte on July 12, 1993. The lowest flow of 12 cfs was recorded on August 7, 1989. Assuming that the annual parameters and distribution given earlier in this problem are true at all times, find the probability that the flow of the Platte exceeds 38,500 cfs or is less than 12 cfs. c) Suppose that biologists determine that the habitat needs of Whooping Cranes require the flow of the river to exceed 2,500 cfs at least 90% of the time during the month of March. Assume that the water flow variance is the same in March as at all times of the year. What is the average water flow that water managers would need to target in March to insure the proper maintenance of the Whooping Cranes’ habitat? * Homework #5 will not be collected. The included problems are designed to reinforce concepts from recent lectures and to assist in exam preparation. Answers for this assignment will be furnished on Tuesday October 8th.2 d) Again, assume that biologists determine that the habitat needs of Whooping Cranes require the flow of the river to exceed 2,500 cfs at least 90% of the time for the month of March. While the extent of their findings may vary, most non-partisan climatologists also believe that global warming exists. Some of these scientists believe that an important and often overlooked aspect of the debate on global warming is related to its effect on the variance of weather. For example, not only have temperatures appeared to increase during the industrial age at rates not observed previously, but the variation in these temperatures has also appeared to increase. Suppose that one of the effects of this change in climate is to increase the variance in rainfall and thereby increase the variance in water flow in the Platte River. If the standard deviation of the Platte River’s flow increases by 25 percent, and water managers set the average water flow in March to 3,700 cfs, will the Whooping Cranes habitat be maintained? (Show your work). 2. Let X be a continuous random variable with PDF 23()2f x x for 11x   a) Verify that 23()2f x x for 11x   satisfies the requirements for or a PDF. b) Derive the expression for the CDF c) Find the probability that X is between -0.5 and 0.25. d) Compute the expected value of X e) Compute the variance of X 3. Fun with Chi-squared. a) Determine the value that separates the bottom 99% and top 1% of a chi-squared distribution with 50 degrees of freedom. b) Is 30 in the bottom 5% of a chi-square distribution 55 degrees of freedom? Explain. c) Let (4,9),XN (10,36),YN and (20,25).WN Use ,X ,Y and W to create a chi-squared distributed random variable. Do you need to make an assumptions about ,X ,Y and W to create the variable in question? If so, what assumptions do you need to make? How many degrees of freedom does your chi-squared random variable have? d) Find the probability that the random variable that you created in part c. is less than 0.216. 4. Professor Wallace has office hours on Friday’s from 10-noon. The number of students visiting Professor Wallace’s office hours has a Poisson distribution with mean 0.5 students per hour. a) What is the probability that exactly 2 students visit Professor Wallace’s office hours during the first hour? b) What is the probability that Professor Wallace gets 2 visitors to office hours in 20 minutes? c) What is the probability no students visit Professor Wallace’s office hours in a given week? d) What is the probability that Professor Wallace goes 30 minutes without a visitor? e) What is the probability that Professor Wallace waits less than 45 minutes for a visitor?3 5. Drug-Sniffing Dogs (aka Fortunate Patients, aka Beer or Mead). At the border between the United States and Mexico, United States drug enforcement agents use dogs that are specially trained to identify the presence of narcotics in the automobiles crossing the border. Suppose that there are 15 dogs at a specific border