Example 1. Consider tossing a coin n times. If X is the number of heads obtained, X is a random variable.
|
|
- Hillary Dean
- 7 years ago
- Views:
Transcription
1 7 Random variables A random variable is a real-valued function defined on some sample space. associates to each elementary outcome in the sample space a numerical value. That is, it Example 1. Consider tossing a coin n times. If X is the number of heads obtained, X is a random variable. Example 2. Consider a stock price which moves each day either up one unit or down one unit, and suppose its initial value is 10$. Let T be the first time the value of the stock hits either 0$ or 20$. Then T is a random variable. Example 3. The lifetime T of a lightbulb is a random variable. In the last example, if we can measure time with infinite precision, then the possible values of T are the non-negative real numbers [0, ). This is an uncountable set: there is no way to enumerate [0, ) in a sequence. We will have to treat random variables of this type separately from the random variables which take values in a countable set. While in practice time can only be measured up to finite precision and consequently the possible values of T will in fact be countable, it is still more convenient mathematically to make the idealization that all values in [0, ) are possible, and we will do so. 7.1 Distribution functions For a random variable X, we can associate the distribution function F X ( ), sometimes called the cumulative distribution function, defined by F X (t) = P (X t). (1) Notice that F X ( ) is defined on all real numbers. The distribution function determines the probability that X falls in an interval: P (a < X b) = P (X b) P (X a) = F X (b) F X (a). Example 4. Suppose a coin with probability p of landing heads is tossed until the first time a heads appears. Let T be the number of tosses required. For a real number t, let [t] denote the integer part of t. We have P (T > t) = P (T > [t]) = P (first [t] tosses are tails ) = (1 p) [t]. 1
2 Consequently, We can use this to compute F T (t) = P (T t) = { 1 (1 p) [t] if t 0, 0 if t < 0. P (2 < T 5) = 1 (1 p) 5 [ 1 (1 p) 2] = (1 p) 2 (1 p) 5. We may want to find the probability that X falls in a closed interval. To this end, we need the following: Proposition 1. Let X be a random variable with distribution function F. Then P (X < t) = lim s t F (s). The value lim s t F (s) is called the left limit of F at t, and is denoted sometime by F (t ). Part of the conclusion of Proposition 1 is that a distribution function has left limits everywhere. To prove this, we first need to show that a probability P ( ) obeys a certain kind of continuity. Lemma 2. (i) Let A 1 A 2 A 3 be a non-decreasing sequence of events. Then lim P (A n) = P ( A k ). (ii) Let A 1 A 2 A 3 be a non-increasing sequence of events. Then lim P (A n) = P ( A k ). Proof. We prove (i). The proof of (ii) is obtained by looking at complements and using (i), and is left to the reader as an exercise. Define B k = A k \ A k 1. The events {B k } are disjoint, A n = n B k, and A k = B k. Thus, P ( A k ) = P ( B k ) = P (B k ) = lim n P (B k ) = lim P ( n B k ) = lim P (A n ). 2
3 Proof of Proposition 1. Notice that {X < t} = {X t 1 }. Then applying Lemma k 2 gives P (X < t) = lim P (X t 1 n ) = lim F (t 1 n ) = lim s t F (s). Thus, we can also use the distribution function of X to calculate other probabilities involving X: P (a X b) = P (X b) P (X < a) = F (b) F (a ) P (X = a) = P (X a) P (X < a) = F (a) F (a ). (2) A random variable X is proper if P ( < X < ) = 1. Almost all random variables we will encounter will be proper, but it is worth noting that there do exist random variables which are not proper. Example 5. Suppose a particle moves on the integer {..., 3, 2, 1, 0, 1, 2, 3,...} as follows: at each move, it moves up one integer with probability 2/3, and moves down one integer with probability 1/3. The particle starts at 0. Let T be the first time that the particle is at 1. The event that the particle never hits 1 is {T = }. We will see later that P (T < ) < 1, so that T is not a proper random variable. Writing {X < } = {X < k}, if X is a finite random variable then applying Lemma 2 again shows Also, 1 = P (X < ) = lim P (X n) = lim F X (n). 0 = P (X < ) = lim P (X n) = lim F X( n) = lim F X(n). m n Finally, since {X t} = {X t + 1 }, using part (ii) of Lemma 2 shows that k lim s t F (s) = F (t), and so F is right-continuous everywhere. We summarize the properties of the distribution function of a random variable X as follows: Proposition 3. Let X be a proper random variable with distribution function F. Then (i) F is right-continuous: lim s t F (s) = F (t), (ii) the left-limits of F exist everywhere, 3
4 (iii) lim t F (t) = 1. (iv) lim t F (t) = 0. The first two properties imply that the worst behavior possible of a distribution function is that it jumps. 8 Discrete random variables We call a random variable which can take on only countable many values a discrete random variable. 8.1 Probability mass functions Let X be a discrete random variable which takes values in the set A = {a 0, a 1, a 2,...}. Associated to X is the function p X ( ), defined by p X (a) = P (X = a). (3) The function p X ( ) is defined for all real numbers, although it will be strictly positive only for a in the set A. A function p( ) satisfying (i) p(a) 0 for all a, (ii) a p(a) = 1 is called a probability mass function, or pmf for short. It is easily checked that p X ( ) satisfies these conditions, and we call it the pmf of X. We write X p( ) to indicate that X has pmf p( ). Notice that from (2) we have p X (a) = P (X = a) = F X (a) F X (a ), (4) so the pmf of X can be determined if the distribution function of X is known. Example 6. Suppose n independent experiments are performed, each of which can result in either success or failure, and suppose that the probability of success on each experiment is p. Such a sequence of experiments is called Bernoulli trials. Let X be the number of successes in these n experiments. The event {X = k} contains all outcomes containing 4
5 exactly k successes and n k failures. There are ( n k) such outcomes, each having probability p k (1 p) n k (by independence.) Thus ( ) n p X (k) = P (X = k) = p k (1 p) n k. (5) k A random variable X having the pmf in (5) is called a Binomial(n, p) random variable, and we write X Binomial(n, p). 8.2 The distribution of a discrete random variable An event determined by X is an event of the form {X A}, where A is a subset of the real numbers. We can find the probability of any event determined by X using only the pmf of X: P (X A) = a A p X (a). (6) Applying (6) to the set A = (, t] gives F X (t) = P (X t) = a A p X (a). (7) Thus the distribution function of X can be computed from the pmf of X. To summarize, we record the following: Proposition 4. Let X be a discrete random variable. Each of the following can be computed using any of the others: (i) The probabilities of all events determined by X, that is, the collection of probabilities {P (X A) : A R}, (ii) the pmf p X ( ) of X, (iii) the distribution function F X ( ) of X. Proof. This is the content of the combination of equations (4), (6), and (7). The collection of probabilities {P (X A) : A R} is called simply the distribution of X. It contains all the probabilistic information about the random variable X. Proposition 4 says that for a discrete random variable, it is enough to specify either the pmf or the distribution function to specify the distribution. Thus, if one is asked to determine the distribution of X, it is sufficient to provide either the pmf or the distribution function. 5
6 9 Continuous random variables A probability density function (abbreviated pdf or sometimes simply density) is a realvalued function f defined on the real numbers satisfying (i) f(t) 0 for all real numbers t, (ii) f(t)dt = 1. A continuous random variable is a random variable X for which there exists a pdf f X so that P (a < X b) = b a f X (t)dt for all a < b. (8) In fact, if (8) holds, then for any subset of real numbers A such that f(t)dt is defined, A the identity P (X A) = f(t)dt (9) is valid. Applying (9) to the set (, t] shows that A F X (t) = P (X t) = t f(s)ds, (10) and so the distribution function of X can be determined from the density function of X. Note that a consequence of (10) is that F X is a continuous function for a continuous random variable, and in particular P (X = a) = F X (a) F X (a ) = 0. Applying the Fundamental Theorem of Calculus to (10) shows that d dt F X(t) = f X (t), (11) at all points t where f X is continuous. Thus if f X is piecewise continuous, as will be the case in this course, then it can be determined from the distribution function via (11). The following summarizes the situation for continuous random variables with piecewise continuous densities: 6
7 Proposition 5. Let X be a continuous random variable with piecewise continuous density. Each of the following can be computed using any of the others: (i) The probabilities of all events determined by X, that is, the collection of probabilities {P (X A) : A R such that f(t)dt is defined}, A (ii) the pdf f X ( ) of X, (iii) the distribution function F X ( ) of X. Proof. This is what equations (9), (10), (11) say. 9.1 Interpretation of density function What is the interpretation of the density function? Suppose that X has a density f, which is continuous at the point a. We have P (a X a + ) = F (a + ) F (a) The right-hand side tends to F (a) = f(a) as 0. Thus we can write that. P (a X a + ) = f(a) + ε 0 ( ), where ε 0 ( ) 0 as 0. Multiplying both sides by, we have that P (a X a + ) = f(a) + ε 0 ( ). }{{} ε( ) If ε( ) = ε 0 ( ), then ε( )/ 0 as 0. Thus we can write P (a X a + ) f(a), (12) where the error in the approximation is ε( ) and satisfies ε( )/ 0 as 0. Equation (12) is useful in interpreting the meaning of a probability density function: the probability of X falling in a very small interval near a is approximated by f(a), where is the length of the interval. 7
8 10 Expected Value Let X be a discrete random variable with the following pmf 2 if a = 1, 5 2 if a = 0, 5 p X (a) = 1 if a = 1, 5 0 if a { 1, 0, 1}. How should the average value of X be defined? A first attempt might be to say that the average value should be 0, since 0 is in the center of the three possible values { 1, 0, 1}. But this does not take into account that X does not assume these values with equal probability. The average should account for not just the values taken on by X, but also the probabilities associated to each of these values. This leads to the definition of the expectation of X, which is a weighted average of the values of X, the weights determined by the pmf or pdf. Precisely, we define E(X) = { a ap X(a) if X is discrete tf X(t)dt if X is continuous. (13) E(X) is only defined when the sum or integral in (13) converges absolutely, that is, we need { a a p X(a) < if X is discrete In the example above, t f X(t)dt < if X is continuous E(X) = = 1 5. We use the terms expected value, mean, and moment all to refer to the expectation of X. Example 7. Let X be a Binomial(n, p) random variable. This means that X has a pmf given by ( ) n p X (k) = p k (1 p) n k, k 8
9 for k = 0, 1,..., n. The pmf is 0 for any other values. Then n ( ) n E(X) = k p k (1 p) n k k k=0 n = k n! k! (n k)!pk (1 p) n k = n n! (k 1)!(n k)! pk (1 p) n k k 1 = np n n 1 (n 1)! (k 1)!(n 1 (k 1))! pk 1 (1 p) n 1 (k 1) (n 1)! = np p k (1 p) n 1 k k!(n 1 k)! k=0 }{{} = np. ( n 1 k ) } {{ } pmf of Binomial(n 1, p) r.v. Example 8. We say that X is an Exponential random variable with parameter λ if it has a pdf { 1 λ f(t) = e 1 λ t if t 0, 0 if t < 0. X has the property that P (X > t + s X > t) = P (X > t). (The reader should check that!) The expected value is the integral E(X) = We can evaluate this by integration by parts: Set 0 t 1 λ e 1 λ t dt. u = t du = dt v = e 1 λ t dv = 1 λ e 1 λ t dt 9
10 so that 0 t 1 1 λ e λ t dt = te 1 λ t + 0 = λe 1 λ t = λ 0 0 e 1 λ t dt Functions of random variables If g : R R is a function, and X is a random variable, then Y = g(x) is a new random variable. To calculate E(Y ) according to the definition (13), we need the pmf if X is discrete, or the pdf if X is continuous. Fortunately, the following proposition tells us how to compute E(Y ) without finding its density or pmf. Proposition 6. Let X be a random variable, and g a real-valued function. { a E(g(X)) = g(a)p X(a) if X is discrete with pmf p X, g(t)f X(t)dt if X is continuous with pdf f X. Proof. We prove the case where X is discrete: E(g(X)) = b = b = b = b = a bp (g(x) = b) b P (X = a) a : g(a)=b a : g(a)=b a : g(a)=b bp X (a) g(a)p X (a). g(a)p X (a) An immediate corollary is the following: 10
11 Corollary 7. Let X be a random variable, and let α and β be constants. Then E(αX + β) = αe(x) + β. Proof. We write the continuous case, the discrete case is similar: Applying Proposition 6 to g(x) = αx + β gives E(αX + β) = (αt + β)f X (t)dt = α tf X (t) + β f X (t)dt = αe(x) + β. 11 Variance Expectation measures the center of mass of a density or pmf. Variance is a measure of how spread out the density or pmf of X is. The random variable Y = (X E(X)) 2 gives the squared distance of X to its mean value. This measures how far X is from its center of mass. Taking the expectation of Y gives the variance of X: { V (X) = E(X E(X)) 2 a = (a E(X))2 p X (a) if X is discrete with pmf p X, (t E(X))2 f X (t)dt if X is continuous with pdf p X. The following is a useful way to compute variance Proposition 8. for a random variable X, V (X) = E(X 2 ) [E(X)] 2. Proof. The proof is similar in the continuous and discrete cases, we show here the discrete 11
12 case: V (X) = a = a (a E(X)) 2 p X (a) (a 2 2aE(X) + [E(X)] 2 )p X (a) = a a 2 p X (a) 2E(X) a ap X (a) + [E(X)] 2 a p X (a) = E(X 2 ) 2E(X)E(X) + [E(X)] 2 = E(X 2 ) [E(X)] 2 12
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
More informationMASSACHUSETTS 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
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationChapter 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,
More informationRandom 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
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationFor a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )
Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll
More information5. 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
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationThe 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
More informationLecture 6: Discrete & Continuous Probability and Random Variables
Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September
More informationSection 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,
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationIEOR 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
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability
CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationSTAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE
STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More informationMathematics of Life Contingencies MATH 3281
Mathematics of Life Contingencies MATH 3281 Life annuities contracts Edward Furman Department of Mathematics and Statistics York University February 13, 2012 Edward Furman Mathematics of Life Contingencies
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information1. (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
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationM2S1 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
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More informationMULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
More informatione.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
More informationSTAT 830 Convergence in Distribution
STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31
More information1 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
More informationProbability 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
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More informationStatistics 100A Homework 4 Solutions
Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationProbability 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
More informationProbability 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,
More informationOverview 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?
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected
More informationLectures 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,
More informationThe 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
More informationA review of the portions of probability useful for understanding experimental design and analysis.
Chapter 3 Review of Probability A review of the portions of probability useful for understanding experimental design and analysis. The material in this section is intended as a review of the topic of probability
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationSection 6.1 Joint Distribution Functions
Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function
More informationRANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. DISCRETE RANDOM VARIABLES.. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a finite or countable
More informationTransformations and Expectations of random variables
Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationLecture 13: Martingales
Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of
More informationExample. A casino offers the following bets (the fairest bets in the casino!) 1 You get $0 (i.e., you can walk away)
: Three bets Math 45 Introduction to Probability Lecture 5 Kenneth Harris aharri@umich.edu Department of Mathematics University of Michigan February, 009. A casino offers the following bets (the fairest
More informationTwo Fundamental Theorems about the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationWald 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,
More information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationNotes on Probability Theory
Notes on Probability Theory Christopher King Department of Mathematics Northeastern University July 31, 2009 Abstract These notes are intended to give a solid introduction to Probability Theory with a
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationPractice problems for Homework 11 - Point Estimation
Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationGambling Systems and Multiplication-Invariant Measures
Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More information1. 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
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding
More informationCITY 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
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More information1.1 Introduction, and Review of Probability Theory... 3. 1.1.1 Random Variable, Range, Types of Random Variables... 3. 1.1.2 CDF, PDF, Quantiles...
MATH4427 Notebook 1 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 1 MATH4427 Notebook 1 3 1.1 Introduction, and Review of Probability
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationMathematical 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.............................
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and
More informationSection 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
More information1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
More informationCardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.
Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationLecture 3: Continuous distributions, expected value & mean, variance, the normal distribution
Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ
More informationMathematics 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
More informationUndergraduate 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
More informationProbability and statistics; Rehearsal for pattern recognition
Probability and statistics; Rehearsal for pattern recognition Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception
More informationContinued 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
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationNonparametric 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
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions
Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationarxiv: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
More informationStatistics 100A Homework 8 Solutions
Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the one-half
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More information