ENSC 1103 Environmental Science (Assignment 7)

Answer the following questions associated with Chapter 15 & 16. You will need to be complete with your answers to receive full credit.

1. Define water pollution, point source, and nonpoint source pollution. Which of the two (point source or nonpoint source) is easier to identify? Which is easier to legislate? Which currently poses the greatest threat to freshwater?

2. Research and describe in your own words the process of lake stratification and lake turnover. (include the layered zones epilimnion, hypolimnion, and thermocline and explain what’s happening and why and how this happens) “When do these events happen in Oklahoma? (This information is not in your book but you will see it again on the exam).

3. Describe the different types of freshwater and marine wetlands. Explain the importance of wetlands.

4. What is meant by the “Great Pacific Garbage Patch”? What type of trash is the biggest concern in the Pacific Gyre? Why? Discuss three ways in which people are fighting pollution in the oceans and our coasts.

5. Describe five major forms of water pollution and provide an example of each. List three examples of measurements that scientists use to determine water quality.

Math 464 - Fall 13 - Homework 8

1. X and Y are independent random variables, each of which has the stan- dard normal distribution. Show that Z = X/Y has a Cauchy distribution.

2. Let X be a standard normal random variable, and let Y = σX + µ where σ > 0. (a) Show that the pdf of Y is normal with mean µ and variance σ2. (b) Show that moment generating function of X is exp(t2/2). (c) Show that the mgf of Y is given by the formula on the formula sheet. (Hint: recall the proposition from class about the mgf of aX +b. This should take almost no computation.)

3. (Exposition) In class we stated a theorem that says that if X and Y are independent continuous random variables and g and h are functions from R to R, then g(X) and h(Y ) are independent random variables. We only proved it for the special case that g and h are increasing functions. In this problem you prove for two more special cases. (a) Prove that if X and Y are independent then X2 and Y 2 are independent. (b) Prove that if X and Y are independent then X and −Y are independent.

4. The Laplace distribution is

f(x) = 1

2 λe−λ|x|, −∞ < x < ∞

where λ > 0 is a parameter. Compute the moment generating function and use it to find the mean and variance.

5. Let X and Y be independent random variables. They each have the exponential distribution with the same λ. Let Z = Y − X. The goal of this problem is to find the density of Z using moment generating functions. (There should be very little computation in your solution.) (a) Find the mgf of −X. Hint: think of −X as (−1)X and recall the propo- sition from class about the mgf of aX + b. (b) Use the fact that −X and Y are independent (which you proved in a previous problem) to find the mgf of Z. (c) Find the density of Z. Hint: don’t compute - find a RV with the same moment generating function.

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