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### CourseNotesMath407(5) (1).56

Course: MATH 407, Spring 2012
School: USC
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Word Count: 351

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{(1 (ii) 1 1 , 2 ) : Y (2 ) = 2} = x=0 {(1 , 2 ) : X (1 ) = x, Y (2 ) = 2} . Problem 7. Suppose X and Y are two random variables, and let f (x, y ) be their joint probability density given as X=1 Y=1 Y=2 1/3 0 X=2 1/3 . 1/3 Compute the marginal density for X and Y . Problem 8. Let X be a discrete random variable that takes the value 0 with probability 1/2 and -1 with probability 1/4 and 1 with...

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{(1 (ii) 1 1 , 2 ) : Y (2 ) = 2} = x=0 {(1 , 2 ) : X (1 ) = x, Y (2 ) = 2} . Problem 7. Suppose X and Y are two random variables, and let f (x, y ) be their joint probability density given as X=1 Y=1 Y=2 1/3 0 X=2 1/3 . 1/3 Compute the marginal density for X and Y . Problem 8. Let X be a discrete random variable that takes the value 0 with probability 1/2 and -1 with probability 1/4 and 1 with probability 1/4. Show that the random variable Y = X 2 is the indicator function for the event that a fair coin lands heads in 1 toss. Problem 9. Let be the space of outcomes obtained when a fair 6 faced dice is thrown 2 times. Let X be the difference between the face in the rst toss and the second toss. Let Y be the sum of the faces in the two tosses. Compute (i) f , the joint density function for (X, Y ) (ii) fX , the marginal density for X (iii) fY , the marginal density for Y (iv) The set of outcomes for which X > 0 and Y 4. Problem 10. Let X be a random variable and let the cdf of X be given by x<0 0 0.22 0 x < 1 1 0.23 x < 2 F (x) = 0.5 2 x<3 0.9 3 x < 4 1 x 4. Compute the pdf for X . Is X a continuous or a discrete random variable? Problem 11*. (Optional, and not for the faint-of-heart.) This problem is very, very hard, and will not be graded; it explores a connection between combinatorics, analysis and probability. If you are interested in studying probability beyond this course, you may attempt it; however, not attempting it will not hinder your ability to follow the material in this course. Some of the material in the problem is a preview of the kind of things that you might do in the rst few weeks of a graduate course in analysis. Let = P (N) = {0, 1}N = {innite sequences consisting of 0 and 1} and let P () be the collection of all subsets of . Show that the following sequence of assertions is true. (a) Let A . Show that A can be thought of as the binary expansion of a real number in the interval [0, 1], and conversely. (b) Use the previous part to show that there is a 1-1, onto, function : [0, 1]. 2
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USC - MATH - 407
(c) Let us dene a probability on [0, 1] by asking that for any interval I = (a, b) P (I ) = b a. This suggeststhat we can dene a probability Q on by asking that for any subset A P (), Q(A) = P (A).There are problems with this denition, but for now, play
USC - MATH - 407
Homework 4 Hints: Due on Febuary 13, 2012.Not all problems in this homework will be graded. The grader will randomly pick some problems tograde, and all the problems are equally likely to be picked for grading . Starred* problems are harder.Problem 1.
USC - MATH - 407
write down all possible unions for sets in B3 , together with the sets in B3 to get a collection B4 . Keepdoing this. Stop when your collections dont pick up any sets that you hadnt seen before.You will notice that you stop picking up new sets in your c
USC - MATH - 407
Homework 5: Due on Febuary 24, 2012.Not all problems in this homework will be graded. The grader will randomly pick some problems tograde, and all the problems are equally likely to be picked for grading . Starred* problems are harder.Problem 1. Let (,
USC - MATH - 407
(d)alima 2axf (x)dxDoes not exist.This distribution is called the standard Cauchy distribution. The average does not exist for this distribution, and neither do higher moments. We will encounter this distribution again in a few weeks.2
USC - MATH - 407
Homework 6: Due on March 05, 2012.Not all problems in this homework will be graded. The grader will randomly pick some problems tograde, and all the problems are equally likely to be picked for grading . Starred* problems are harder.Problem 1. Let X an
USC - MATH - 407
Homework 7: Due on March 26, 2012.Not all problems in this homework will be graded. The grader will randomly pick some problems tograde, and all the problems are equally likely to be picked for grading . Starred* problems are harder.Problem 1. Page 217
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Texas Woman's University - ENG - 1021
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Texas Woman's University - ENG - 1021
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Texas Woman's University - ENG - 1021
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Tallahassee Community College should adopt a smoke-free campus for many reasons. Ifor one to not accept people choosing to take the risk of getting cancer when I pay taxesthat go towards there health care. How can the government allow the distribution o
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