# 3. Renewal Theory. Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times.

Save this PDF as:

Size: px
Start display at page:

Download "3. Renewal Theory. Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times."

## Transcription

1 3. Renewal Theory Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times. Set S n = n i=1 X i and N(t) = max {n : S n t}. Then {N(t)} is a renewal process. Set µ = E[X n ]. We assume µ > 0 (i.e., F(0)<1) but we allow the possibility µ =. Many questions about more complex processes can be addressed by identifying a relevant renewal process. Therefore, we study renewal theory in some detail. The renewal function is m(t) = E[N(t)]. Writing the c.d.f. of S n as F n = F F... F, the n-fold convolution of F, we have P[N(t) = n] = P[S n t] P[S n+1 t] = F n (t) F n+1 (t). 1

2 Recall: if X and Y are independent, with c.d.f. F X and F Y respectively, then F X+Y (z) = F X (z y)df Y (y) = F Y (z x)df X (x). This identity is called a convolution, written F X+Y = F X F Y = F Y F X. Taking Laplace transforms gives F X+Y = F X FY. It follows that m(t) = E[N(t)] = = F n (t) n=1 n ( F n (t) F n+1 (t) ) n=1 2

3 Limit Properties for Renewal Processes (L1) lim n S n /n = µ with probability 1. Why? (L2) lim t N(t) = with probability 1. Why? 3

4 (L3) lim t N(t) t = 1 µ with probability 1. Why? 4

5 (L4) The Elementary Renewal Theorem lim t m(t) t = 1 µ Why is L4 different from L3? Because it is not always [ true that, ] for any stochastic ] process, E = lim t E. lim t N(t) t counter-example [ N(t) t 5

6 Exchanging Expectation and Summation If X 1, X 2,... are non-negative random variables, E [ n=1 X n] = n=1 E[X n]. For any X 1, X 2,... with n=1 E[ X n ] <, E[ n=1 X n] = n=1 E[X n]. These results do not require independence; they are consequences of two basic theorems. (checking this is an optional exercise.) Monotone Convergence Theorem: If Y 1, Y 2,... is an increasing sequence of random variables (i.e., Y n Y n+1 ) then E [ lim n Y n ] = limn E[Y n ]. Dominated Convergence Theorem: For any Y 1, Y 2,..., if there is a random variable Z with E[Z] < and Y n < Z for all n, then E [ lim n Y n ] = limn E[Y n ]. 6

7 Stopping Times A non-negative integer-valued random variable N is a stopping time for a sequence X 1, X 2,... if E[N] < and {N = n} is determined by X 1,...,X n. Example: If cards are dealt successively from a deck, the position of the first spade is a stopping time, say N. Note that N 1, the position of the last card before a spade appears, is not a stopping time (but N + 1 is). Example: Any gambling strategy cannot depend on future events! Thus, the decision to stop gambling must be a stopping time. If X 1, X 2,..., are independent, then {N = n} is independent of X n+1, X n+2,... if N is a stopping time. (This is the definition in Ross, p. 104, but we will use the more general definition, which will be needed later). 7

8 Example For a renewal process, N(t) + 1 is a stopping time for the interarrival sequence X 1, X 2,... (Is N(t) a stopping time? Why.) (i) Show {N(t) + 1 = n} is determined by X 1,...,X n (ii) Show that E[N(t)] <, i.e., check that m(t) < 8

9 Wald s Equation: If X 1, X 2,... are iid with E[ X 1 ] < and N is a stopping time, then [ N ] E = E[N] E[X 1 ] Proof n=1 X n 9

10 Proof of the elementary Renewal Theorem (i) Apply Wald s equation to the stopping time N(t) + 1, to get lim inf t m(t) t 1 µ note that we use the definition lim inf t f(t) = lim t {inf s>t f(s)} 10

11 m(t) (ii) Show lim sup t t 1 µ by applying Wald s equation to a truncated renewal process, with X n if X n M X n = M if X n > M and S n = n X { i=1 i, Ñ(t) = sup n : S } n t, µ = E [ X], m(t) = E [Ñ(t) ]. 11

12 Example. Customers arrive at a bank as a Poisson process, rate λ. There is only one clerk, and service times are iid with distribution G. Customers only enter the bank if the clerk is available; if the clerk is busy they go away without entering. (In queue terminology, this is an M/G/1 queue with no buffer.) (a) Identify a relevant renewal process. (b) At what rate do customers enter the bank? (what does rate mean here?) 12

13 . (c) What fraction of potential customers enter the bank? (d) What fraction of time is the server busy? 13

14 (L5) Central Limit Theorem for N(t). Suppose µ = E[X 1 ] < and σ 2 = Var(X 1 ) <. Then, [ ] N(t) t/µ lim t P < y = 1 y /2 2π dx e x2 σ t/µ 3 This is a consequence of the central limit theorem for S n (see Ross, Theorem for a proof). Why is there a µ 3 in the denominator? We can give a dimension analysis, as a check: 14

15 A diagram and an informal argument can suggest a method of proof, and help explain the result: 15

16 (L6) The Key Renewal Theorem lim t t 0 h(t x) dm(x) = 1 µ Under the requirements that 0 h(t) dt (i) F is not lattice, i.e. there is no d 0 such that n=0 P[ X 1 =nd ] = 1. (ii) h(t) is directly Riemann integrable (see following slide). L6 can be seen as a result on sums over a moving window: suppose at time t we consider a sum H t with contribution h(t x) for an event at time t x. Then, H t = N(t) 1 n=0 h(t S N(t) n ) = t 0 h(t x)dn(x). Exchanging integration and expectation, we get E[H t ] = t 0 h(t x) E[dN(x)] = t 0 h(t x) dm(x). 16

17 Example: Set h(t) = 1 for t a 0 else. Then H t counts the number of events in the time window [t a, t]. In this case, since h(t) is directly Riemann integrable, (L6) gives the following result: (L7) Blackwell s theorem, part (i) If F is not lattice, then, for a > 0, lim t m(t) m(t a) = a/µ How does this follow? 17

18 If F is lattice, then they key renewal theorem fails: counter-example... A modification of L(7)(i) still holds: (L7) Blackwell s theorem, part (ii) If F is lattice with period d then lim n E [ number of renewals at nd ] = d µ Note that the number of renewals at nd can be written as dn(nd). 18

19 Definition: h : [0, ] R is directly Riemann integrable (dri) if lim a sup h(t) a 0 n=1 (n 1)a t na = lim a 0 a n=1 inf h(t). (n 1)a t na i.e., if the limits of upper and lower bounds exist and are equal, then h is dri and the integral is equal to this limit. h : [0, T] R is Riemann integrable (Ri) if T/a lim a a 0 n=1 sup h(t) (n 1)a t na T/a = lim a 0 a n=1 inf h(t). (n 1)a t na and h : [0, ] R is Riemann integrable if the limit T h(t) dt = lim 0 T h(t) dt exists. 0 The dri definition requires the ability to take a limit (i.e., an infinite sum) inside the lim a 0. This infinite sum does not always exist for a Riemann integrable function. 19

20 It is possible to find an h(t) which is Ri but not dri and for which the key renewal theorem fails. (e.g., Feller, An Introduction to Probability Theory and its Applications, Vol. II ). If h(t) is dri then h(t) is Ri, and the two integrals are equal. If h(t) is Ri and h(t) 0 and h(t) is non-increasing then it can be shown that h(t) is dri (this is sometimes an easy way to show dri). 20

21 Example (Alternating Renewal Process) Each interarrival time X n consists of an on-time Z n followed by an off-time Y n. Suppose (Y n, Z n ) are iid, but Y n and Z n may be dependent. Suppose Y n and Z n have marginal distributions G and H and X n = Y n + Z n F. X n if X n x E.g., let Z n = x if X n > x. Then the alternating process is on if t S N(t) x. The quantity A(t) = t S N(t) is the age of the renewal process at time t. X n if X n x E.g., let Y n = x if X n > x. In this case, the process is off if S N(t)+1 t x. This alternating process is appropriate to study the residual life Y (t) = S N(t)+1 t. 21

22 (L8) Alternating Renewal Theorem If E[Z 1 + Y 1 ] < and F is non-lattice, then lim P[ system is on at t ] = t E[Z 1 ] E[Z 1 ] + E[Y 1 ] Proof. The method is to condition on S N(t) and apply the key renewal theorem. This is an example of a powerful conditioning approach. 22

23 Lemma: df SN(t) (s) = F(t s) dm(s), s > 0 F(t) + F(t) dm(0), s = 0 Proof of lemma. 23

24 Proof of (L8) continued 24

25 Example (Regenerative Process) A stochastic process {X(t), t 0} is regenerative if there are random times S 1, S 2,... at which the process restarts. This means that {X(t + u), t 0} given S n = u is conditionally independent of {X(t), 0 t u} and has the same distibution as {X(t), t 0}. A regenerative process defines a renewal process with X n =S n S n 1 and N(t) = max{n : S n t}. Each interarrival interval for {N(t)} is called a cycle of {X(t)}. An alternating renewal process is a regenerative process: An alternating renewal process X(t) takes values on and off, and the times at which X(t) switches from off to on are renewal times, by the construction of the alternating renewal process. 25

26 (L9) Regenerative Process Theorem If X(t) is non-lattice, taking values 0, 1, 2,..., and the expected length of a cycle is finite then E[time in state j for one cycle] lim P[X(t)=j] = t E[time of a cycle] Outline of Proof Example: If A(t) and Y (t) are the age and residual life processes of a renewal process N(t), how can (L9) be applied to find lim t P [ Y (t) > x, A(t) > y ]? 26

27 Renewal Reward Processes At each renewal time S n for a renewal process {N(t)} we receive a reward R n. We think of the reward as accumulating over the time interval [S n 1, S n ], so we assume that R n depends on X n but is independent of (R m, X m ) for m n. In other words, (R n, X n ) is an iid sequence of vectors. The reward process is R(t) = N(t) n=1 R n. (L10) If E[R 1 ] < and E[X 1 ] < then (i) R(t) lim t t = E[R 1] E[X 1 ] with probability 1 (ii) lim t E[R(t)] t = E[R 1] E[X 1 ] Outline of proof 27

28 Example: A component is replaced if it reaches the age T years, at a planned replacement cost C 1. If it fails earlier, it is replaced at a failure replacement cost C 2 > C 1. Components have iid lifetimes, with distribution F. Find the long run cost per unit time. Solution 28

29 Delayed Renewal Processes All the limit results for renewal processes still apply when the first arrival time has a different distribution from subsequent interarrival times. This situation arises often because the time t = 0 is the special time at which we start observing a process. Example: Suppose cars on the freeway pass an observer with independent inter-arrival times, i.e., as a renewal process. Unless the observer starts counting when a car has just passed, the time to the first arrival will have a different distribution to subsequent intarrivals [an exception to this is when the interarrival times are memoryless!] Formally, let X 1, X 2,... be independent, with X 1 G, X n F for n 2, and S n = n i=1 X i. Then N D (t) = sup {n : S n t} is a delayed renewal process 29

30 Defining m D (t) = E[N D (t)] and µ = E[X 2 ], (LD1) lim n S n n = µ w.p.1 (LD2) lim t N D (t) = w.p.1 (LD3) (LD4) lim t N D (t) t = 1 µ w.p.1 lim t m D (t) t = 1 µ (LD5) [ lim P ND (t) t/µ t σ t/µ 3 ] < y = 1 2π y (LD6) If F is not lattice and h is directly Riemann integrable, lim t t 0 h(t x) dm D(x) = 1 µ (LD7) (i) If F is not lattice, 0 lim t m D (t + a) m D (t) = a/µ e x2/2 dx h(t)dt (ii) If F and G are lattice, period d, lim n E [ number of renewals at nd ] = d/µ 30

31 Example: Suppose a coin lands H with probability p. The coin is tossed until k consecutive heads occur in a row. Show that the expected number of tosses is k i=1 (1/p)i. Solution 31

32 Solution continued 32

### M/M/1 and M/M/m Queueing Systems

M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

### IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

### Exponential Distribution

Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1

### THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

### Mathematical Finance

Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

### Asymptotics of discounted aggregate claims for renewal risk model with risky investment

Appl. Math. J. Chinese Univ. 21, 25(2: 29-216 Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially

### A Note on the Ruin Probability in the Delayed Renewal Risk Model

Southeast Asian Bulletin of Mathematics 2004 28: 1 5 Southeast Asian Bulletin of Mathematics c SEAMS. 2004 A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100

Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,

### The Exponential Distribution

21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

### Jointly Distributed Random Variables

Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Lectures 5 & / Introduction to Queueing Theory

Lectures 5 & 6 6.263/16.37 Introduction to Queueing Theory MIT, LIDS Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS PS PS Packet Network PS PS

### Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

### Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers

Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### e.g. arrival of a customer to a service station or breakdown of a component in some system.

Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be

### Aggregate Loss Models

Aggregate Loss Models Chapter 9 Stat 477 - Loss Models Chapter 9 (Stat 477) Aggregate Loss Models Brian Hartman - BYU 1 / 22 Objectives Objectives Individual risk model Collective risk model Computing

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### 1 IEOR 6711: Notes on the Poisson Process

Copyright c 29 by Karl Sigman 1 IEOR 6711: Notes on the Poisson Process We present here the essentials of the Poisson point process with its many interesting properties. As preliminaries, we first define

### 1. Let X and Y be normed spaces and let T B(X, Y ).

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### The SIS Epidemic Model with Markovian Switching

The SIS Epidemic with Markovian Switching Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH (Joint work with A. Gray, D. Greenhalgh and J. Pan) Outline Motivation 1 Motivation

### Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### 3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk

Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Jose Blanchet and Peter Glynn December, 2003. Let (X n : n 1) be a sequence of independent and identically distributed random

### Lecture Notes 1. Brief Review of Basic Probability

Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very

### 6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory

6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory Massachusetts Institute of Technology Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS

### Sufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Section 5.1 Continuous Random Variables: Introduction

Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

### THE PRIME NUMBER THEOREM

THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description

### Automata and Rational Numbers

Automata and Rational Numbers Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca http://www.cs.uwaterloo.ca/~shallit 1/40 Representations

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### Homework 6 (due November 4, 2009)

Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### 6.041/6.431 Spring 2008 Quiz 2 Wednesday, April 16, 7:30-9:30 PM. SOLUTIONS

6.4/6.43 Spring 28 Quiz 2 Wednesday, April 6, 7:3-9:3 PM. SOLUTIONS Name: Recitation Instructor: TA: 6.4/6.43: Question Part Score Out of 3 all 36 2 a 4 b 5 c 5 d 8 e 5 f 6 3 a 4 b 6 c 6 d 6 e 6 Total

### Lectures on Stochastic Processes. William G. Faris

Lectures on Stochastic Processes William G. Faris November 8, 2001 2 Contents 1 Random walk 7 1.1 Symmetric simple random walk................... 7 1.2 Simple random walk......................... 9 1.3

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### Resource Provisioning and Network Traffic

Resource Provisioning and Network Traffic Network engineering: Feedback traffic control closed-loop control ( adaptive ) small time scale: msec mainly by end systems e.g., congestion control Resource provisioning

### MTH135/STA104: Probability

MTH135/STA14: Probability Homework # 8 Due: Tuesday, Nov 8, 5 Prof Robert Wolpert 1 Define a function f(x, y) on the plane R by { 1/x < y < x < 1 f(x, y) = other x, y a) Show that f(x, y) is a joint probability

### Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T..

Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = {Ω, F,

### Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Insurance models and risk-function premium principle

Insurance models and risk-function premium principle Aditya Challa Supervisor : Prof. Vassili Kolokoltsov July 2, 212 Abstract Insurance sector was developed based on the idea to protect people from random

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### Joint Distribution and Correlation

Joint Distribution and Correlation Michael Ash Lecture 3 Reminder: Start working on the Problem Set Mean and Variance of Linear Functions of an R.V. Linear Function of an R.V. Y = a + bx What are the properties

### L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Poisson Processes. Chapter 5. 5.1 Exponential Distribution. The gamma function is defined by. Γ(α) = t α 1 e t dt, α > 0.

Chapter 5 Poisson Processes 5.1 Exponential Distribution The gamma function is defined by Γ(α) = t α 1 e t dt, α >. Theorem 5.1. The gamma function satisfies the following properties: (a) For each α >

### Week 1: Introduction to Online Learning

Week 1: Introduction to Online Learning 1 Introduction This is written based on Prediction, Learning, and Games (ISBN: 2184189 / -21-8418-9 Cesa-Bianchi, Nicolo; Lugosi, Gabor 1.1 A Gentle Start Consider

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### 6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

### UNIFORM ASYMPTOTICS FOR DISCOUNTED AGGREGATE CLAIMS IN DEPENDENT RISK MODELS

Applied Probability Trust 2 October 2013 UNIFORM ASYMPTOTICS FOR DISCOUNTED AGGREGATE CLAIMS IN DEPENDENT RISK MODELS YANG YANG, Nanjing Audit University, and Southeast University KAIYONG WANG, Southeast

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### M2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung

M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2.

Chapter 6 Brownian Motion 6. Normal Distribution Definition 6... A r.v. X has a normal distribution with mean µ and variance σ, where µ R, and σ > 0, if its density is fx = πσ e x µ σ. The previous definition

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### CONDITIONAL EXPECTATION AND MARTINGALES

CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### Probability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X

Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random

### SPARE PARTS INVENTORY SYSTEMS UNDER AN INCREASING FAILURE RATE DEMAND INTERVAL DISTRIBUTION

SPARE PARS INVENORY SYSEMS UNDER AN INCREASING FAILURE RAE DEMAND INERVAL DISRIBUION Safa Saidane 1, M. Zied Babai 2, M. Salah Aguir 3, Ouajdi Korbaa 4 1 National School of Computer Sciences (unisia),

### Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard