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Course: STAT 532, Spring 2012
School: Purdue
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Word Count: 341

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5 5 Homework - Additional Problems 5.1. (This is exercise 1.8 in Introduction to Stochastic Processes by Lawler). Consider a simple random walk on the graph below. (Recall that simple random walk on a graph is the markov chain which at each time moves to an adjacent vertex, each adjacent vertex having the same probability.) A B C D E a) In the long run, about what fraction of time is spent in vertex A? b)...

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5 5 Homework - Additional Problems 5.1. (This is exercise 1.8 in Introduction to Stochastic Processes by Lawler). Consider a simple random walk on the graph below. (Recall that simple random walk on a graph is the markov chain which at each time moves to an adjacent vertex, each adjacent vertex having the same probability.) A B C D E a) In the long run, about what fraction of time is spent in vertex A? b) Suppose a walker starts in vertex A. What is the expected number of steps until the walker returns to A? c) Suppose a walker starts in vertex C. What is the expected number of visits to B before the walker returns to A? d) Suppose a walker starts in vertex B. What is the probability that the walker reaches A before the walker reaches C? 13 5.2. The following problem completes part of the proof of Theorem 1.23 that we didn't do in class. a) Let z1 , z2 , zn , . . . be a sequence of non-negative such numbers that the limit L = +z limn z1 +z2n n exists and is finite. Prove that this implies that maxkn zk = 0. n n lim Hint: first show that it is enough to prove that limn zn /n = 0. b) Using part a) above, prove the following fact: If Z1 , Z2 , . . . are i.i.d. random variables with E|Z1 | < , then maxkn |Zk | lim = 0. n n 14 5.3. As in the proof of Theorem 1.20 in the book, for x, y I let x (y) = n=0 Px (Xn = y, Tx > n). It was shown that if the state space is irreducible and all states are recurrent then x is a stationary measure for the Markov chain. a) What goes wrong in the proof if one of the states is transient? In particular, let x be a transient state and explain why x is not a stationary measure. b) What goes wrong if the state space is not irreducible? In particular, let x be recurrent and show that x is a stationary measure but that x (y) (0, ) for all y I anymore. / 15
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Purdue - STAT - 532
5Homework 5 - Additional Problems5.1. (This is exercise 1.8 in Introduction to Stochastic Processes by Lawler). Consider a simple random walk on the graph below. (Recall that simple random walk on a graph is the markov chain which at each time moves to
Purdue - STAT - 532
10Homework 10 - Additional Problems10.1. Consider the following modification to the Poisson janitor example (example 3.5 in the book). Suppose that the average lifetime of the lightbulbs is F = 60 and that the janitor comes to check on the bulb accordin
Purdue - STAT - 532
13Homework 13 - Additional Problems13.1. In the standard branching process model individuals reproduce offspring at rate and die at rate (see Example 4.4 in the book). Consider the following modification where there the population can also grow by immig
Purdue - STAT - 532
Stationary distributions of continuous time Markov chainsJonathon Peterson April 13, 2012The following are some notes containing the statement and proof of some theorems I covered in class regarding explicit formulas for the stationary distribution and
Purdue - STAT - 532
1Stopping TimesIn class we defined a stopping time in the following way. T is a stopping time for the Markov chain cfw_Xn n0 if for any n the event cfw_T = n can be expressed in terms of X0 , X1 , . . . , Xn . Lemma 1. There are two other equivalent def
Purdue - MA - 232
Purdue - MA - 232
71. Perform Gauss-Jordan elimination on the following matrix. I am grading yourwork not just your answer so be clear what you are doing and write legibly.(20 pts)12A:L;A--F3+sr4 lZl-e+rz-rl ZOoL,+-f-tI4ilIN0'l- (r+C1at*tt )lzloooLa
Purdue - MA - 232
MA 232Calculus II Group Work Solutions (L4-L5)Fall '101. Find the general form of the integral using integration by parts twice. x2 cos(x) dx Solution: x cos(x) dx = x sin(x) -2 22x sin(x) dx - cos(x) 2 dx]u = x2 dv = cos(x) dx du = 2x dx v = sin(x)
Purdue - MA - 232
Purdue - MA - 232
Purdue - MA - 232
Purdue - MA - 232
Purdue - MA - 232
Purdue - MA - 232
MA 232Calculus II Practice Exam 1Fall '10Show all work to receive full credit. Only approved calculators are allowed on quizzes and exams. Answers which are not fully justified might not receive full credit. The practice exam is a 1 1/2 hour practice e
Purdue - MA - 232
7i:t:u'b=zt cfw_.-3-3v*3 colX=Z t \= -3tl=oP \-/gtzxzM*l J= x-! y Lcfw_'n+t)\t'+&amp;_[:ftTot./,?]
Purdue - MA - 232
Practice Exam 3 1. Find x and y. 2 3 0 y 2 5 + = 5 1 x -1 y 22. Find x and y. 2 1 3 2 x -1 1 0 = y 2 0 13. Solve the differential equations. (a)dy dx= 3xy;y(2) = e3 .(b) y + 2y = 2x;y(0) = 3 . 24. Sketch the vector field of solutions to the auto
Purdue - MA - 232
MA 232Calculus IIFall 10Practice Quiz 1 (L1)Show work and have the right answer for full credit. Wrong answer with some right work isworth partial credit, and right answer with wrong work is worth partial credit.1. Find the area between the curves f
Purdue - MA - 232
MA 232Calculus II Practice Quiz 2 (L3)Fall '10Show work and have the right answer for full credit. Wrong answer with some right work is worth partial credit, and right answer with wrong work is worth partial credit. 1. Find the value of the given integ
Purdue - MA - 232
MA 232Calculus IIFall 10Practice Quiz 3 (L5)Show work and have the right answer for full credit. Wrong answer with some right work isworth partial credit, and right answer with wrong work is worth partial credit.1. Find the integral two ways. First,
Purdue - MA - 232
MA 232Calculus II Practice Quiz 4 (L7)Fall '10Show work and have the right answer for full credit. Wrong answer with some right work is worth partial credit, and right answer with wrong work is worth partial credit. 1. Find the area of a sphere of radi
Purdue - MA - 232
MA 232Calculus IIFall 10Practice Quiz 5 (L11)Show work and have the right answer for full credit. Wrong answer with some right work isworth partial credit, and right answer with wrong work is worth partial credit.For each given function f (x, y ), n
Purdue - MA - 232
MA 232Calculus IIFall 10Quiz 1: ANSWERS (L1)Show work and have the right answer for full credit. Wrong answer with some right work isworth partial credit, and right answer with wrong work is worth partial credit.1. Find the area underf (x) =exx1
Purdue - MA - 232
MA 232Calculus II Quiz 1 (L1)Fall '10Show work and have the right answer for full credit. Wrong answer with some right work is worth partial credit, and right answer with wrong work is worth partial credit. 1. Find the area under f (x) = and over the i
Purdue - MA - 232
MA 232Calculus IIFall 10Quiz 2:ANSWERS (L3)1. Find F () ifF () = cos sin2 and F (0) = 2/3 .Solution:By the Fundamental Theorem of Calculus, we can nd F () byintegrating F () = F () d. So,F () ==cos sin2 du = sin u2 dudu = cos d= 1/3 u3 + C
Purdue - MA - 232
MA 232Calculus II Quiz 2 (L3)Fall '10Show work and have the right answer for full credit. Wrong answer with some right work is worth partial credit, and right answer with wrong work is worth partial credit. 1. Find F () if F () = cos sin2 and F (0) = 2
Purdue - MA - 232
MA 232Calculus IIFall 10Quiz 3:ANSWERS (L5)Show work and have the right answer for full credit. Wrong answer with some right work isworth partial credit, and right answer with wrong work is worth partial credit.1. Find the value of the integral.5l
Purdue - MA - 232
MA 23200: Calculus for the Life Sciences II Fall 2010COURSE WEBPAGE: http:/www.math.purdue.edu/ma232 REQUIRED TEXTBOOK: Calculus for the Life Sciences, Marvin L. Bittinger, Neal Brand and John Quintanilla, 2006 PREREQUISITE: MA 23100 (or: MA 29000 Sectio
Purdue - MA - 220
MA 220 Exam 2 Answers, Spring 2012Problem1)2)Form AAForm BCActual Answer5ABx 5, 0, 2, 3, and 6 only(Problem 2 will be a free problem, however. All answers will be counted.)3)BE15 x 24)BDm 485)AA56)BEy 3x 67)BD\$13.808)DE
Purdue - MA - 220
MA 22000Final Exam Practice Problems1. If f (x) = -x2 - 3x + 4, calculate f (-2). A. -6 B. 0 C. 2 D. 6 E. 14 2. If f (x) = 2x2 - x + 1, find and simplify f (x + 2). A. 2x2 - x + 3 B. 2x2 + 7x + 7 C. 2x2 - x + 7 D. 2x2 + 7x + 11 E. 2x2 - x + 11 3. Simpli
Purdue - MA - 220
Even Answers for Chapter 2 Problems (2nd half of text) Section 2.1 6) 50) 54) slope is4 38)slope is1 410)slope is -3differentiable at all x except -3 and 3, where there are cusps (, 3) (3,3) (3, ) differentiable at all x except -2 and 2, where the
Purdue - MA - 220
Even Answers: Chapter 1 (2nd half of textbook)Section 1.336)y 2 x 2 or 2 x y 2 038)y x 3 or x y 3 040)y 1 or y 1 042)x246)y1679xor 16 x 15x 79 0151554)y 2 x 6 or 2 x y 6 080)a)86)V 30000t 825000, 0 t 2588)3014 students90)a)c)S
Purdue - MA - 220
MA 22000, Lesson 26 Notes Section 3.1 A function is increasing if its function values (y's) are rising as x values get larger. A function is decreasing if its function values (y's) are falling as x values get larger. Formal Definition: A function f is inc
Purdue - MA - 220
MA 22000 Lesson 27 Notes (2nd half of text, section 3.2) Relative Extrema, Absolute Extrema in an interval In the last lesson, we found intervals hwere a function was increasing or intervals where that function was decreasing. At a point, where a function
Purdue - MA - 220
MA 220 Lesson 28 Notes Section 3.3 (p. 191, 2nd half of text) The property of the graph of a function `curving' upward or downward is defined as the concavity of the graph of a function. Concavity if how the derivative (that describes if a function is dec
Purdue - MA - 220
MA 220 Lesson 29, Section 3.7 (part 1) Curve Sketching: 1) Find (if possible) any y-intercept or x-intercepts. To find y-intercept, let x = 0 and solve. To find x-intercept(s), let y = 0 and solve. (Finding x-intercept(s) is not always be easy to accompli
Purdue - MA - 220
MA 220 Lesson 30, Section 3.7 (part 2)Curve Sketching:1)Find (if possible) any y-intercept or x-intercepts. To find y-intercept, let x = 0 andsolve. To find x-intercept(s), let y = 0 and solve. (Finding x-intercept(s) is not always beeasy to accompli
Purdue - MA - 220
MA 22000 Lesson 31 Notes Section 3.4 Optimization Problems An optimization problems involves finding a value that would determine a maximum or minimum for a problem. For many of these problems, you will have to write two equations initially. The primary e
Purdue - MA - 220
MA 22000 Lesson 32 Notes Review of Guidelines for Optimization Problems (finding maximums or minimums). An optimization problems involves finding a value that would determine a maximum or minimum for a problem. For many of these problems, you will have to
Purdue - MA - 220
MA 22000 Lesson 33 Notes 3 part of section 3.4 and section 3.5rdWe will continue to use the same guidelines for finding maximum or minimum values as in the previous two lessons. Example 1: A rectangular page is to contain 64 square inches of print. The
Purdue - MA - 220
MA 22000 Lesson 37 Notes Section 4.3 (part 1), Pages 273 280 Derivative of the NATURAL EXPONENTIAL Functiond x dy [e ] e x or if y f ( x) e x , y ex dx dx d u dy du du [e ] eu or if y f (u ) eu , euu or eu cx dx dx dx (using chain rule)Example 1: Find t
Purdue - MA - 220
MA 22000 Lesson 39 Lesson Notes(2 half of text) Section 4.4, Logarithmic FunctionsndDefinition of General Logarithmic Function:A logarithmic function, denoted by y logb x , is equivalent to b y x .In previous algebra classes, you may have often used
Purdue - MA - 220
MA 22000 Lesson 40 Notes Section 4.5 (part 1) Derivative of the natural logarithmic function:d 1 [ln x] dx x argument.)(In words, the derivative of a natural logarithmic function is the reciprocal of theLet u be a function of x, then d 1 1 du [ln u ] u
Purdue - MA - 220
MA 22000 Lesson 41 Notes Section 4.5 (2nd half of text) (Continuation of derivatives of logarithmic functions) Reminder of rules of derivatives of logarithmic functions: Derivative of the natural logarithmic function:d 1 [ln x] dx x(In words, the deriva
Purdue - MA - 220
MA 22000 Lesson 42 Notes Section 4.6, Exponential Growth and Decay Quote from textbook (page 299): &quot;Real-life situations that involve exponential growth and decay deal with a substance or population whose rate of change at any time t is proportional to th
Purdue - MA - 220
MA 22000 Notes, Lesson 36, (2nd half of text) section 4.2Natural Exponential FunctionsSummary: A general and basic exponential function is f ( x) a x , where the base, a is anypositive number except one.Examine the function values for various bases.A
Purdue - MA - 220
MA 22000 Lesson 38 Notes Section 4.3 (2nd half of text) part 2 This is a continuation of finding derivatives of exponential functions. Example 1: Find each derivative.a ) f ( x ) x (e )2 2xb)2e x x2c)e x e x g ( x) 4d ) F ( x) x3e x e xe)f ( x) 2
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