Definition A random variable X is said to have a Weibull distribution with parameters α and β (α > 0, β > 0) if the pdf of X is
|
|
- Nancy Candice Allison
- 5 years ago
- Views:
Transcription
1 Weibull Distribution Definition A random variable X is said to have a Weibull distribution with parameters α and β (α > 0, β > 0) if the pdf of X is { α β x α 1 e f (x; α, β) = α (x/β)α x 0 0 x < 0 Remark: 1. The family of Weibull distributions was introduced by the Swedish physicist Waloddi Weibull in We use X WEB(α, β) to denote that the rv X has a Weibull distribution with parameters α and β. Liang Zhang (UofU) Applied Statistics I July 1, / 15
2 Weibull Distribution Remark: 3. When α = 1, the pdf becomes { 1 β f (x; β) = e x/β x 0 0 x < 0 which is the pdf for an exponential distribution with parameter λ = 1 β. Thus we see that the exponential distribution is a special case of both the gamma and Weibull distributions. 4. There are gamma distributions that are not Weibull distributios and vice versa, so one family is not a subset of the other. Liang Zhang (UofU) Applied Statistics I July 1, / 15
3 Weibull Distribution Liang Zhang (UofU) Applied Statistics I July 1, / 15
4 Weibull Distribution Liang Zhang (UofU) Applied Statistics I July 1, / 15
5 Weibull Distribution Proposition Let X be a random variable such that X WEI(α, β). Then ( E(X ) = βγ ) α ( and V (X ) = β {Γ ) [ ( Γ )] } 2 α α The cdf of X is F (x; α, β) = { 1 e (x/β)α x 0 0 x < 0 Liang Zhang (UofU) Applied Statistics I July 1, / 15
6 Weibull Distribution Example: The shear strength (in pounds) of a spot weld is a Weibull distributed random variable, X WEB(400, 2/3). a. Find P(X > 410). b. Find P(X > 410 X > 390). c. Find E(X ) and V (X ). d. Find the 95th percentile. Liang Zhang (UofU) Applied Statistics I July 1, / 15
7 Weibull Distribution In practical situations, γ = min(x ) > 0 and X γ has a Weibull distribution. Example (Problem 74): Let X = the time (in 10 1 weeks) from shipment of a defective product until the customer returns the product. Suppose that the minimum return time is γ = 3.5 and that the excess X 3.5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1.5. a. What is the cdf of X? b. What are the expected return time and variance of return time? c. Compute P(X > 5). d. Compute P(5 X 8). Liang Zhang (UofU) Applied Statistics I July 1, / 15
8 Lognormal Distribution Definition A nonnegative rv X is said to have a lognormal distribution if the rv Y = ln(x ) has a normal distribution. The resulting pdf of a lognormal rv when ln(x ) is normally distributed with parameters µ and σ is { 1 f (x; µ, σ) = 2πσx e [ln(x) µ]2 /(2σ 2 ) x 0 0 x < 0 Remark: 1. We use X LOGN(µ, σ 2 ) to denote that rv X have a lognormal distribution with parameters µ and σ. 2. Notice here that the parameter µ is not the mean and σ 2 is not the variance, i.e. µ E(X ) and σ 2 V (X ) Liang Zhang (UofU) Applied Statistics I July 1, / 15
9 Lognormal Distribution Liang Zhang (UofU) Applied Statistics I July 1, / 15
10 Lognormal Distribution Proposition If X LOGN(µ, σ 2 ), then E(X ) = e µ+σ2 /2 and V (X ) = e 2µ+σ2 (e σ2 1) The cdf of X is F (x; µ, σ) = P(X x) = P[ln(X ) ln(x)] ( = P Z ln(x) µ ) ( ) ln(x) µ = Φ σ σ x 0 where Φ(z) is the cdf of the standard normal rv Z. Liang Zhang (UofU) Applied Statistics I July 1, / 15
11 Lognormal Distribution Example (Problem 115) Let I i be the input current to a transistor and I 0 be the output current. Then the current gain is proportional to ln(i 0 /I i ). Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measurement), so that current gain = X = ln(i 0 /I i ). Assume X is normally distributed with µ = 1 and σ = a. What is the probability that the output current is more than twice the input current? b. What are the expected value and variance of the ratio of output to input current? c. What value r is such that only 5% chance we will have the ratio of output to input current exceed r? Liang Zhang (UofU) Applied Statistics I July 1, / 15
12 Beta Distribution Definition A random variable X is said to have a beta distribution with parameters α, β(both positive), A, and B if the pdf of X is ( α 1 ( β 1 1 f (x; α, β, A, B) = B A Γ(α+β) Γ(α) Γ(β) x A B A) B x B A) A x B 0 otherwise The case A = 0, B = 1 gives the standard beta distribution. Remark: We use X BETA(α, β, A, B) to denote that rv X has a beta distribution with parameters α, β, A, and B. Liang Zhang (UofU) Applied Statistics I July 1, / 15
13 Beta Distribution Proposition If X BETA(α, β, A, B), then E(X ) = A + (B A) α α + β and V (X ) = (B A) 2 αβ (α + β) 2 (α + β + 1) Liang Zhang (UofU) Applied Statistics I July 1, / 15
14 Beta Distribution Liang Zhang (UofU) Applied Statistics I July 1, / 15
15 Beta Distribution Example (Problem 127) An individual s credit score is a number calculated based on that person s credit history which helps a lender determine how much he/she should be loaned or what credit limit should be established for a credit card. An article in the Los Angeles Times gave data which suggested that a beta distribution with parameters A = 150, B = 850, α = 8, β = 2 would provide a reasonable approximation to the distribution of American credit scores. [Note: credit scores are integer-valued]. a. Let X represent a randomly selected American credit score. What are the mean value and standard deviation of this random variable? What is the probability that X is within 1 standard deviation of its mean value? b. What is the approximate probability that a randomly selected score will exceed 750 (which lenders consider a very good score)? Liang Zhang (UofU) Applied Statistics I July 1, / 15
Continuous Random Variables and Probability Distributions
Continuous Random Variables and Probability Distributions Instructor: Lingsong Zhang 1 4.1 Probability Density Functions Probability Density Functions Recall from Chapter 3 that a random variable X is
More information15 Discrete Distributions
Lecture Note 6 Special Distributions (Discrete and Continuous) MIT 4.30 Spring 006 Herman Bennett 5 Discrete Distributions We have already seen the binomial distribution and the uniform distribution. 5.
More informationChapter 5: Generating Random Numbers from Distributions
Chapter 5: Generating Random Numbers from Distributions See Reading Assignment OR441-DrKhalid Nowibet 1 Review 1 Inverse Transform Generate a number u i between 0 and 1 (one U-axis) and then find the corresponding
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 20, 2012 Introduction Introduction The world of the model-builder
More informationGamma and Normal Distribuions
Gamma and Normal Distribuions Sections 5.4 & 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 15-3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationProbability Distributions for Continuous Variables. Probability Distributions for Continuous Variables
Probability Distributions for Continuous Variables Probability Distributions for Continuous Variables Let X = lake depth at a randomly chosen point on lake surface If we draw the histogram so that the
More informationApplied Statistics I
Applied Statistics I Liang Zhang Department of Mathematics, University of Utah July 8, 2008 Liang Zhang (UofU) Applied Statistics I July 8, 2008 1 / 15 Distribution for Sample Mean Liang Zhang (UofU) Applied
More informationReview 1: STAT Mark Carpenter, Ph.D. Professor of Statistics Department of Mathematics and Statistics. August 25, 2015
Review : STAT 36 Mark Carpenter, Ph.D. Professor of Statistics Department of Mathematics and Statistics August 25, 25 Support of a Random Variable The support of a random variable, which is usually denoted
More informationProbability Distributions for Discrete RV
An example: Assume we toss a coin 3 times and record the outcomes. Let X i be a random variable defined by { 1, if the i th outcome is Head; X i = 0, if the i th outcome is Tail; Let X be the random variable
More information3 Continuous Random Variables
Jinguo Lian Math437 Notes January 15, 016 3 Continuous Random Variables Remember that discrete random variables can take only a countable number of possible values. On the other hand, a continuous random
More informationExpected Values, Exponential and Gamma Distributions
Expected Values, Exponential and Gamma Distributions Sections 5.2-5.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 14-3339 Cathy Poliak,
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.
UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Christopher Barr University of California, Los Angeles,
More informationContinuous random variables
Continuous random variables Can take on an uncountably infinite number of values Any value within an interval over which the variable is definied has some probability of occuring This is different from
More informationClassical and Bayesian inference
Classical and Bayesian inference AMS 132 January 18, 2018 Claudia Wehrhahn (UCSC) Classical and Bayesian inference January 18, 2018 1 / 9 Sampling from a Bernoulli Distribution Theorem (Beta-Bernoulli
More informationApplied Statistics I
Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 17, 2008 Liang Zhang (UofU) Applied Statistics I June 17, 2008 1 / 22 Random Variables Definition A dicrete random variable
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationTest Problems for Probability Theory ,
1 Test Problems for Probability Theory 01-06-16, 010-1-14 1. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30
More informationExponential, Gamma and Normal Distribuions
Exponential, Gamma and Normal Distribuions Sections 5.4, 5.5 & 6.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy Poliak,
More informationProbability Density Functions
Probability Density Functions Probability Density Functions Definition Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that
More informationContinuous random variables and probability distributions
and probability distributions Sta. 113 Chapter 4 of Devore March 12, 2010 Table of contents 1 2 Mathematical definition Definition A random variable X is continuous if its set of possible values is an
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( )
UCLA STAT 35 Applied Computational and Interactive Probability Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Chris Barr Continuous Random Variables and Probability
More informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 29, 2014 Introduction Introduction The world of the model-builder
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
More informationTopic 4: Continuous random variables
Topic 4: Continuous random variables Course 3, 216 Page Continuous random variables Definition (Continuous random variable): An r.v. X has a continuous distribution if there exists a non-negative function
More informationClosed book and notes. 60 minutes. Cover page and four pages of exam. No calculators.
IE 230 Seat # Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators. Score Exam #3a, Spring 2002 Schmeiser Closed book and notes. 60 minutes. 1. True or false. (for each,
More informationLecture 17: The Exponential and Some Related Distributions
Lecture 7: The Exponential and Some Related Distributions. Definition Definition: A continuous random variable X is said to have the exponential distribution with parameter if the density of X is e x if
More informationMATH : EXAM 2 INFO/LOGISTICS/ADVICE
MATH 3342-004: EXAM 2 INFO/LOGISTICS/ADVICE INFO: WHEN: Friday (03/11) at 10:00am DURATION: 50 mins PROBLEM COUNT: Appropriate for a 50-min exam BONUS COUNT: At least one TOPICS CANDIDATE FOR THE EXAM:
More informationTopic 4: Continuous random variables
Topic 4: Continuous random variables Course 003, 2018 Page 0 Continuous random variables Definition (Continuous random variable): An r.v. X has a continuous distribution if there exists a non-negative
More information1 Probability and Random Variables
1 Probability and Random Variables The models that you have seen thus far are deterministic models. For any time t, there is a unique solution X(t). On the other hand, stochastic models will result in
More informationChapter 3 Common Families of Distributions
Lecture 9 on BST 631: Statistical Theory I Kui Zhang, 9/3/8 and 9/5/8 Review for the previous lecture Definition: Several commonly used discrete distributions, including discrete uniform, hypergeometric,
More informationMath438 Actuarial Probability
Math438 Actuarial Probability Jinguo Lian Department of Math and Stats Jan. 22, 2016 Continuous Random Variables-Part I: Definition A random variable X is continuous if its set of possible values is an
More informationChing-Han Hsu, BMES, National Tsing Hua University c 2015 by Ching-Han Hsu, Ph.D., BMIR Lab. = a + b 2. b a. x a b a = 12
Lecture 5 Continuous Random Variables BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 215 by Ching-Han Hsu, Ph.D., BMIR Lab 5.1 1 Uniform Distribution
More informationReview for the previous lecture
Lecture 1 and 13 on BST 631: Statistical Theory I Kui Zhang, 09/8/006 Review for the previous lecture Definition: Several discrete distributions, including discrete uniform, hypergeometric, Bernoulli,
More informationCommon ontinuous random variables
Common ontinuous random variables CE 311S Earlier, we saw a number of distribution families Binomial Negative binomial Hypergeometric Poisson These were useful because they represented common situations:
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More informationRandom variables, Expectation, Mean and Variance. Slides are adapted from STAT414 course at PennState
Random variables, Expectation, Mean and Variance Slides are adapted from STAT414 course at PennState https://onlinecourses.science.psu.edu/stat414/ Random variable Definition. Given a random experiment
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationSlides 8: Statistical Models in Simulation
Slides 8: Statistical Models in Simulation Purpose and Overview The world the model-builder sees is probabilistic rather than deterministic: Some statistical model might well describe the variations. An
More informationBe sure that your work gives a clear indication of reasoning. Use notation and terminology correctly.
MATH 232 Fall 2009 Test 1 Name: Instructions. Be sure that your work gives a clear indication of reasoning. Use notation and terminology correctly. No mystry numbers: If you use sage, Mathematica, or your
More informationCHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS. 6.2 Normal Distribution. 6.1 Continuous Uniform Distribution
CHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Recall that a continuous random variable X is a random variable that takes all values in an interval or a set of intervals. The distribution of a continuous
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Exercise 4.1 Let X be a random variable with p(x)
More informationLet X be a continuous random variable, < X < f(x) is the so called probability density function (pdf) if
University of California, Los Angeles Department of Statistics Statistics 1A Instructor: Nicolas Christou Continuous probability distributions Let X be a continuous random variable, < X < f(x) is the so
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationARCONES MANUAL FOR THE SOA EXAM P/CAS EXAM 1, PROBABILITY, SPRING 2010 EDITION.
A self published manuscript ARCONES MANUAL FOR THE SOA EXAM P/CAS EXAM 1, PROBABILITY, SPRING 21 EDITION. M I G U E L A R C O N E S Miguel A. Arcones, Ph. D. c 28. All rights reserved. Author Miguel A.
More informationLecture 4. Continuous Random Variables and Transformations of Random Variables
Math 408 - Mathematical Statistics Lecture 4. Continuous Random Variables and Transformations of Random Variables January 25, 2013 Konstantin Zuev (USC) Math 408, Lecture 4 January 25, 2013 1 / 13 Agenda
More informationContinuous Probability Spaces
Continuous Probability Spaces Ω is not countable. Outcomes can be any real number or part of an interval of R, e.g. heights, weights and lifetimes. Can not assign probabilities to each outcome and add
More informationContinuous random variables
Continuous random variables Continuous r.v. s take an uncountably infinite number of possible values. Examples: Heights of people Weights of apples Diameters of bolts Life lengths of light-bulbs We cannot
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The Moment-Generating Function Technique
More informationProbability Distributions Columns (a) through (d)
Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationChapter Learning Objectives. Probability Distributions and Probability Density Functions. Continuous Random Variables
Chapter 4: Continuous Random Variables and Probability s 4-1 Continuous Random Variables 4-2 Probability s and Probability Density Functions 4-3 Cumulative Functions 4-4 Mean and Variance of a Continuous
More informationBMIR Lecture Series on Probability and Statistics Fall, 2015 Uniform Distribution
Lecture #5 BMIR Lecture Series on Probability and Statistics Fall, 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University s 5.1 Definition ( ) A continuous random
More informationX = lifetime of transistor
Kelley Problem 4-57 /3 Suppose that when a transistor of a certain type is subjected to an accelerated life test, the lifetime, X (in weeks) has a gamma distribution with mean 24 weeks and standard deviation
More information3 Modeling Process Quality
3 Modeling Process Quality 3.1 Introduction Section 3.1 contains basic numerical and graphical methods. familiar with these methods. It is assumed the student is Goal: Review several discrete and continuous
More informationStatistics 427: Sample Final Exam
Statistics 427: Sample Final Exam Instructions: The following sample exam was given several quarters ago in Stat 427. The same topics were covered in the class that year. This sample exam is meant to be
More informationDefinition 1.1 (Parametric family of distributions) A parametric distribution is a set of distribution functions, each of which is determined by speci
Definition 1.1 (Parametric family of distributions) A parametric distribution is a set of distribution functions, each of which is determined by specifying one or more values called parameters. The number
More informationMidterm Examination. STA 205: Probability and Measure Theory. Thursday, 2010 Oct 21, 11:40-12:55 pm
Midterm Examination STA 205: Probability and Measure Theory Thursday, 2010 Oct 21, 11:40-12:55 pm This is a closed-book examination. You may use a single sheet of prepared notes, if you wish, but you may
More informationChapter 4: CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Chapter 4: CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Gamma Distribution Weibull Distribution Lognormal Distribution Sections 4-9 through 4-11 Another exponential distribution example
More informationChapter 4 - Lecture 3 The Normal Distribution
Chapter 4 - Lecture 3 The October 28th, 2009 Chapter 4 - Lecture 3 The Standard Chapter 4 - Lecture 3 The Standard Normal distribution is a statistical unicorn It is the most important distribution in
More informationp. 6-1 Continuous Random Variables p. 6-2
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability (>). Often, there is interest in random variables
More informationChapter 2 Continuous Distributions
Chapter Continuous Distributions Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationProbability Distributions
Probability Distributions Series of events Previously we have been discussing the probabilities associated with a single event: Observing a 1 on a single roll of a die Observing a K with a single card
More informationApplied Statistics and Probability for Engineers. Sixth Edition. Chapter 4 Continuous Random Variables and Probability Distributions.
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 4 Continuous Random Variables and Probability Distributions 4 Continuous CHAPTER OUTLINE Random
More informationChapter 4 Continuous Random Variables and Probability Distributions
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 4 Continuous Random Variables and Probability Distributions 4 Continuous CHAPTER OUTLINE 4-1
More informationMathematical statistics
October 4 th, 2018 Lecture 12: Information Where are we? Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 1: Continuous Random Variables Section 4.1 Continuous Random Variables Section 4.2 Probability Distributions & Probability Density
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More informationChapter 3.3 Continuous distributions
Chapter 3.3 Continuous distributions In this section we study several continuous distributions and their properties. Here are a few, classified by their support S X. There are of course many, many more.
More informationIntroduction to Probability and Statistics Slides 3 Chapter 3
Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan
More informationMathematical statistics
October 1 st, 2018 Lecture 11: Sufficient statistic Where are we? Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
More informationContinuous Distributions
Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution (Continuous) 1.10.4-5 Exponential and Gamma Distributions: Distance between crossovers Prof. Tesler Math 283 Fall
More informationMath 3339 Homework 6 (Sections )
Math 3339 Homework 6 (Sections 5. 5.4) Name: Key PeopleSoft ID: Instructions: Homework will NOT be accepted through email or in person. Homework must be submitted through CourseWare BEFORE the deadline.
More informationOutline PMF, CDF and PDF Mean, Variance and Percentiles Some Common Distributions. Week 5 Random Variables and Their Distributions
Week 5 Random Variables and Their Distributions Week 5 Objectives This week we give more general definitions of mean value, variance and percentiles, and introduce the first probability models for discrete
More information0, otherwise, (a) Find the value of c that makes this a valid pdf. (b) Find P (Y < 5) and P (Y 5). (c) Find the mean death time.
1. In a toxicology experiment, Y denotes the death time (in minutes) for a single rat treated with a toxin. The probability density function (pdf) for Y is given by cye y/4, y > 0 (a) Find the value of
More informationCOMPSCI 240: Reasoning Under Uncertainty
COMPSCI 240: Reasoning Under Uncertainty Andrew Lan and Nic Herndon University of Massachusetts at Amherst Spring 2019 Lecture 20: Central limit theorem & The strong law of large numbers Markov and Chebyshev
More informationSTAT 509 Section 3.4: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 509 Section 3.4: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. A continuous random variable is one for which the outcome
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationChapter 3. Julian Chan. June 29, 2012
Chapter 3 Julian Chan June 29, 202 Continuous variables For a continuous random variable X there is an associated density function f(x). It satisifies many of the same properties of discrete random variables
More information18.05 Exam 1. Table of normal probabilities: The last page of the exam contains a table of standard normal cdf values.
Name 18.05 Exam 1 No books or calculators. You may have one 4 6 notecard with any information you like on it. 6 problems, 8 pages Use the back side of each page if you need more space. Simplifying expressions:
More informationExponential Distribution and Poisson Process
Exponential Distribution and Poisson Process Stochastic Processes - Lecture Notes Fatih Cavdur to accompany Introduction to Probability Models by Sheldon M. Ross Fall 215 Outline Introduction Exponential
More informationStat410 Probability and Statistics II (F16)
Stat4 Probability and Statistics II (F6 Exponential, Poisson and Gamma Suppose on average every /λ hours, a Stochastic train arrives at the Random station. Further we assume the waiting time between two
More informationS6880 #7. Generate Non-uniform Random Number #1
S6880 #7 Generate Non-uniform Random Number #1 Outline 1 Inversion Method Inversion Method Examples Application to Discrete Distributions Using Inversion Method 2 Composition Method Composition Method
More informationMidterm Examination. STA 205: Probability and Measure Theory. Wednesday, 2009 Mar 18, 2:50-4:05 pm
Midterm Examination STA 205: Probability and Measure Theory Wednesday, 2009 Mar 18, 2:50-4:05 pm This is a closed-book examination. You may use a single sheet of prepared notes, if you wish, but you may
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
More informationQuestion Points Score Total: 76
Math 447 Test 2 March 17, Spring 216 No books, no notes, only SOA-approved calculators. true/false or fill-in-the-blank question. You must show work, unless the question is a Name: Question Points Score
More informationExponential & Gamma Distributions
Exponential & Gamma Distributions Engineering Statistics Section 4.4 Josh Engwer TTU 7 March 26 Josh Engwer (TTU) Exponential & Gamma Distributions 7 March 26 / 2 PART I PART I: EXPONENTIAL DISTRIBUTION
More informationMATH 3200 PROBABILITY AND STATISTICS M3200SP081.1
MATH 3200 PROBABILITY AND STATISTICS M3200SP081.1 This examination has twenty problems, of which most are straightforward modifications of the recommended homework problems. The remaining problems are
More informationChapter 4: Continuous Random Variables and Probability Distributions
Chapter 4: and Probability Distributions Walid Sharabati Purdue University February 14, 2014 Professor Sharabati (Purdue University) Spring 2014 (Slide 1 of 37) Chapter Overview Continuous random variables
More informationContinuous Random Variables
MATH 38 Continuous Random Variables Dr. Neal, WKU Throughout, let Ω be a sample space with a defined probability measure P. Definition. A continuous random variable is a real-valued function X defined
More informationStat751 / CSI771 Midterm October 15, 2015 Solutions, Comments. f(x) = 0 otherwise
Stat751 / CSI771 Midterm October 15, 2015 Solutions, Comments 1. 13pts Consider the beta distribution with PDF Γα+β ΓαΓβ xα 1 1 x β 1 0 x < 1 fx = 0 otherwise, for fixed constants 0 < α, β. Now, assume
More informationSome worked exercises from Ch 4: Ex 10 p 151
Some worked exercises from Ch 4: Ex 0 p 5 a) It looks like this density depends on two parameters how are we going to plot it for general k and θ? But note that we can re-write the density as a constant
More informationJoint p.d.f. and Independent Random Variables
1 Joint p.d.f. and Independent Random Variables Let X and Y be two discrete r.v. s and let R be the corresponding space of X and Y. The joint p.d.f. of X = x and Y = y, denoted by f(x, y) = P(X = x, Y
More informationCH5 CH6(Sections 1 through 5) Homework Problems
550.40 CH5 CH6(Sections 1 through 5) Homework Problems 1. Part of HW #6: CH 5 P1. Let X be a random variable with probability density function f(x) = c(1 x ) 1 < x < 1 (a) What is the value of c? (b) What
More informationLecture 5: Moment generating functions
Lecture 5: Moment generating functions Definition 2.3.6. The moment generating function (mgf) of a random variable X is { x e tx f M X (t) = E(e tx X (x) if X has a pmf ) = etx f X (x)dx if X has a pdf
More informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationMath May 13, Final Exam
Math 447 - May 13, 2013 - Final Exam Name: Read these instructions carefully: The points assigned are not meant to be a guide to the difficulty of the problems. If the question is multiple choice, there
More information