STA 4321/5325 Solution to Homework 5 March 3, 2017
|
|
|
- Lesley Moody
- 6 years ago
- Views:
Transcription
1 STA 4/55 Solution to Homework 5 March, 7. Suppose X is a RV with E(X and V (X 4. Find E(X +. By the formula, V (X E(X E (X E(X V (X + E (X. Therefore, in the current setting, E(X V (X + E (X Therefore, E(X + E(X + 4X + 4 E(X + 4E(X (WMS, Problem 4.8. The proportion of time per day that all checkout counters in a supermarket are busy is a RV Y with PDF { cy ( y 4, y f(y, elsewhere. (a Find the value of c that makes f(y a probability density function. (b Find E(Y. (a Observe that f(x dx Hence, from c f(x dx, we get (b Note that E(Y y ( y 4 dy ( z z 4 ( dz (substitute z y ( b a ( z z 4 dz g(x dx ( z + z z 4 dz (z 4 z 5 + z 6 dz ( z 5 5 z6 6 + z7 7 yf(y dy c 5. a y y ( y 4 dy b g(x dx ( ( z z 4 dz (substitute z y ( z + z z z 4 dz (z 4 z 5 + z 6 z 7 dz
2 ( z z6 6 ( z7 7 z An absolutely continuous RV X with PDF f is called symmetric (about, if for every x R, f(x f( x. (a Assuming it exists, show that E(X. (b If F denotes the DF of X, show that F (. Hence find the median of X. (You can assume F to be strictly increasing on a neighborhood of. [Hint: Use the properties of odd and even functions.] (a Method (Direct proof: Note that E(X xf(x dx ( yf( y( dy ( yf( y dy ( yf(y dy yf(y dy E(X Hence, E(X E(X. substitue y x dy dx ( b a g(x dx g(x dx a since f(y f( y for all y b Method (Using property of odd functions: By definition, E(X xf(x dx. Note that the integrand g(x xf(x is odd, since g( x xf( x xf(x g(x. Hence, E(X, an (definite integral of an odd function over the symmetric interval (,, is zero. (b We have, f(y dy (integral of a PDF over the entire real line. integrand f(y is an even function since f(y f( y. Therefore, f(y dy Thus, F ( P (Y P (Y >. f(y dy P (Y > P (Y >. Note that the To find the median φ /, first note that F is strictly increasing on a neighborhood of. So, F exists as a function on that neighborhood. Therefore, φ / is obtained by solving F (φ / ( φ / F F (F (.
3 4. (WMS, Problem 4.. The proportion of time Y that an industrial robot is in operation during a 4-hour week is a random variable with probability density function { y, y f(y, elsewhere. (a Find E(Y and V (Y. (b For the robot under study, the profit X for a week is given by X Y 6. Find E(X and V (X. (c Find an interval in which the profit should lie for at least 75% of the weeks that the robot is in use. and (a We have E(Y E(Y yf(y dy y f(y dy y y dy y y y dy y4 4 Therefore, V (Y E(Y E (Y (b By the formula, E(X E(Y 6 E(Y 6 (/ 6 / and V (X V (Y 6 V (Y /9. (c Recall from Chebyshev s theorem, the two standard deviation (k interval about mean has probability greater than or equal to k %. Hence, required interval in the current setting [/ /9, /+ /9] [.9476, 67.64]. 5. (WMS, Problem The failure of a circuit board interrupts work that utilizes a computing system until a new board is delivered. The delivery time, Y, is uniformly distributed on the interval one to five days. The cost of a board failure and interruption includes the fixed cost c of a new board and a cost that increases proportionally to Y. If C is the cost incurred, C + c Y. (c is a constant. (a Find the probability that the delivery time exceeds two days. (b In terms of c and c, find the expected cost associated with a single failed circuit board. (a By assumption, Y U(θ, θ 5. Let F Y ( denote the DF of Y. Therefore, required probability P (Y > P (Y F Y ( θ θ θ (b Recall, if Y U(θ, θ then E(Y θ +θ and V (Y (θ θ 4. Therefore, E(Y V (Y + E (Y (See solution to problem. Hence, E(C E(c + c Y + c E(Y + c.
4 6. Recall that the MGF of a RV X is defined by M X (t : E(e tx. Hence, for continuous X, M X (t etx f(x dx, f being the PDF of X. From this definition, find the MGF of X when X Exp(β. Hence (by differentiating find E(X and E(X. Then find V (X. By definition, M X (t e tx f(x dx provided β t > t < β. Note that M X(t d dt M X(t β β β e tx β e x β dx e e ( β tx dx ( e βt x β dx ( βt β ( βt β x ( βt ( β β ( βt M X(t d dt M X(t d dt M X(t β βt, ( βt ( β β ( βt. Therefore, E(X M X (t t β and E(X M X (t t β. Hence V (X E(X E (X β β β. 7. (WMS, Problem A manufacturing plant uses a specific bulk product. The amount of product used in one day can be modeled by an exponential distribution with β 4 (measurements in tons. (a Find the probability that the plant will use more than 4 tons on a given day. (b How much of the bulk product should be stocked so that the plants chance of running out of the product is only.5? Let X denote the amount (in tons of product used in one day. Then by assumption X Exp(β 4. So, the DF of X is F X (x e x/β e x/4 for x. (a Required probability P (X > 4 P (X 4 F X (4 ( e 4/4 e.679. (b We need x (in tons such that P (X > x.5. Now P (X > x F X (x e x/4. Therefore, e x/4.5 x/4 log(.5 x 4 log( tons. 8. (WMS, Problem 4.9. Let Y have an exponential distribution with P (Y >.8. Find E(Y and P (Y.7. 4
5 Let Y Exp(β. We need to find β. Recall Y has DF F Y (y e y/β for y. Therefore, P (Y > F Y ( e /β.8 β log(.8 β log(.8.8. Hence, E(Y β.8 and P (Y.7 e.7/
Contents 1. Contents
Contents 1 Contents 6 Distributions of Functions of Random Variables 2 6.1 Transformation of Discrete r.v.s............. 3 6.2 Method of Distribution Functions............. 6 6.3 Method of Transformations................
1 Solution to Problem 2.1
Solution to Problem 2. I incorrectly worked this exercise instead of 2.2, so I decided to include the solution anyway. a) We have X Y /3, which is a - function. It maps the interval, ) where X lives) onto
[Chapter 6. Functions of Random Variables]
![[Chapter 6. Functions of Random Variables]](/thumbs/92/109648703.jpg "[Chapter 6. Functions of Random Variables]")
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
Chapter 4. Chapter 4 sections
Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation
Lecture 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
STA 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)
Review 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
STAT515, Review Worksheet for Midterm 2 Spring 2019
STAT55, Review Worksheet for Midterm 2 Spring 29. During a week, the proportion of time X that a machine is down for maintenance or repair has the following probability density function: 2( x, x, f(x The
Random Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
Brief 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
1 Expectation of a continuously distributed random variable
OCTOBER 3, 204 LECTURE 9 EXPECTATION OF A CONTINUOUSLY DISTRIBUTED RANDOM VARIABLE, DISTRIBUTION FUNCTION AND CHANGE-OF-VARIABLE TECHNIQUES Expectation of a continuously distributed random variable Recall
E[X n ]= dn dt n M X(t). ). What is the mgf? Solution. Found this the other day in the Kernel matching exercise: 1 M X (t) =
![E[X n ]= dn dt n M X(t). ). What is the mgf? Solution. Found this the other day in the Kernel matching exercise: 1 M X (t) =](/thumbs/94/121808901.jpg "E[X n ]= dn dt n M X(t). ). What is the mgf? Solution. Found this the other day in the Kernel matching exercise: 1 M X (t) =")
Chapter 7 Generating functions Definition 7.. Let X be a random variable. The moment generating function is given by M X (t) =E[e tx ], provided that the expectation exists for t in some neighborhood of
Closed 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,
4.1 Expected value of a discrete random variable. Suppose we shall win $i if the die lands on i. What is our ideal average winnings per toss?
Chapter 4 Mathematical Expectation 4.1 Expected value of a discrete random variable Examples: (a) Given 1, 2, 3, 4, 5, 6. What is the average? (b) Toss an unbalanced die 1 times. Lands on 1 2 3 4 5 6 Probab..1.3.1.2.2.1
Math 3215 Intro. Probability & Statistics Summer 14. Homework 5: Due 7/3/14
Math 325 Intro. Probability & Statistics Summer Homework 5: Due 7/3/. Let X and Y be continuous random variables with joint/marginal p.d.f. s f(x, y) 2, x y, f (x) 2( x), x, f 2 (y) 2y, y. Find the conditional
Solution to Assignment 3
The Chinese University of Hong Kong ENGG3D: Probability and Statistics for Engineers 5-6 Term Solution to Assignment 3 Hongyang Li, Francis Due: 3:pm, March Release Date: March 8, 6 Dear students, The
Stat 515 Midterm Examination II April 4, 2016 (7:00 p.m. - 9:00 p.m.)
Name: Section: Stat 515 Midterm Examination II April 4, 2016 (7:00 p.m. - 9:00 p.m.) The total score is 120 points. Instructions: There are 10 questions. Please circle 8 problems below that you want to
Mathematics 426 Robert Gross Homework 9 Answers
Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX
p. 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
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.
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
1. If X has density. cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. f(x) =
1. If X has density f(x) = { cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. 2. Let X have density f(x) = { xe x, 0 < x < 0, otherwise. (a) Find P (X > 2). (b) Find
Question 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
Distributions 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
STA 584 Supplementary Examples (not to be graded) Fall, 2003
Page 1 of 8 Central Michigan University Department of Mathematics STA 584 Supplementary Examples (not to be graded) Fall, 003 1. (a) If A and B are independent events, P(A) =.40 and P(B) =.70, find (i)
0, 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
Continuous 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
Limiting Distributions
We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the two fundamental results
Continuous Distributions
A normal distribution and other density functions involving exponential forms play the most important role in probability and statistics. They are related in a certain way, as summarized in a diagram later
Ch3 Operations on one random variable-expectation
Ch3 Operations on one random variable-expectation Previously we define a random variable as a mapping from the sample space to the real line We will now introduce some operations on the random variable.
Homework 10 (due December 2, 2009)
Homework (due December, 9) Problem. Let X and Y be independent binomial random variables with parameters (n, p) and (n, p) respectively. Prove that X + Y is a binomial random variable with parameters (n
P (x). all other X j =x j. If X is a continuous random vector (see p.172), then the marginal distributions of X i are: f(x)dx 1 dx n
JOINT DENSITIES - RANDOM VECTORS - REVIEW Joint densities describe probability distributions of a random vector X: an n-dimensional vector of random variables, ie, X = (X 1,, X n ), where all X is are
Stat410 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
Regression and Statistical Inference
Regression and Statistical Inference Walid Mnif wmnif@uwo.ca Department of Applied Mathematics The University of Western Ontario, London, Canada 1 Elements of Probability 2 Elements of Probability CDF&PDF
4.1 The Expectation of a Random Variable
STAT 42 Lecture Notes 93 4. The Expectation of a Random Variable This chapter begins the discussion of properties of random variables. The focus of this chapter is on expectations of random variables.
ECE353: Probability and Random Processes. Lecture 7 -Continuous Random Variable
ECE353: Probability and Random Processes Lecture 7 -Continuous Random Variable Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu Continuous
Errata for the ASM Study Manual for Exam P, Fourth Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA
Errata for the ASM Study Manual for Exam P, Fourth Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA (krzysio@krzysio.net) Effective July 5, 3, only the latest edition of this manual will have its
Notes for Math 324, Part 19
48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which
BMIR Lecture Series on Probability and Statistics Fall 2015 Discrete RVs
Lecture #7 BMIR Lecture Series on Probability and Statistics Fall 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University 7.1 Function of Single Variable Theorem
Constructing Approximations to Functions
Constructing Approximations to Functions Given a function, f, if is often useful to it is often useful to approximate it by nicer functions. For example give a continuous function, f, it can be useful
Question 1. Question 4. Question 2. Question 5. Question 3. Question 6.
İstanbul Kültür University Faculty of Engineering MCB17 Introduction to Probability Statistics Second Midterm Fall 21-21 Number: Name: Department: Section: Directions You have 9 minutes to complete the
Chapter 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
Math 362, Problem set 1
Math 6, roblem set Due //. (4..8) Determine the mean variance of the mean X of a rom sample of size 9 from a distribution having pdf f(x) = 4x, < x
Stat 5101 Notes: Algorithms
Stat 5101 Notes: Algorithms Charles J. Geyer January 22, 2016 Contents 1 Calculating an Expectation or a Probability 3 1.1 From a PMF........................... 3 1.2 From a PDF...........................
Chapter 6: Functions of Random Variables
Chapter 6: Functions of Random Variables We are often interested in a function of one or several random variables, U(Y 1,..., Y n ). We will study three methods for determining the distribution of a function
4 Moment generating functions
4 Moment generating functions Moment generating functions (mgf) are a very powerful computational tool. They make certain computations much shorter. However, they are only a computational tool. The mgf
Stat 366 A1 (Fall 2006) Midterm Solutions (October 23) page 1
Stat 366 A1 Fall 6) Midterm Solutions October 3) page 1 1. The opening prices per share Y 1 and Y measured in dollars) of two similar stocks are independent random variables, each with a density function
Exam P Review Sheet. for a > 0. ln(a) i=0 ari = a. (1 r) 2. (Note that the A i s form a partition)
Exam P Review Sheet log b (b x ) = x log b (y k ) = k log b (y) log b (y) = ln(y) ln(b) log b (yz) = log b (y) + log b (z) log b (y/z) = log b (y) log b (z) ln(e x ) = x e ln(y) = y for y > 0. d dx ax
CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables
CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables Instructor: Erik Sudderth Brown University Computer Science March 5, 2015 Homework Submissions Electronic
YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH A Test #2 June 11, Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 2. A Test #2 June, 2 Solutions. (5 + 5 + 5 pts) The probability of a student in MATH 4 passing a test is.82. Suppose students
Econ 508B: Lecture 5
Econ 508B: Lecture 5 Expectation, MGF and CGF Hongyi Liu Washington University in St. Louis July 31, 2017 Hongyi Liu (Washington University in St. Louis) Math Camp 2017 Stats July 31, 2017 1 / 23 Outline
Chapter 3, 4 Random Variables ENCS Probability and Stochastic Processes. Concordia University
Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University ENCS6161 p.1/47 The Notion of a Random Variable A random variable X is a function that assigns a real
Order Statistics and Distributions
Order Statistics and Distributions 1 Some Preliminary Comments and Ideas In this section we consider a random sample X 1, X 2,..., X n common continuous distribution function F and probability density
Week 2. Review of Probability, Random Variables and Univariate Distributions
Week 2 Review of Probability, Random Variables and Univariate Distributions Probability Probability Probability Motivation What use is Probability Theory? Probability models Basis for statistical inference
Math 426: Probability MWF 1pm, Gasson 310 Exam 3 SOLUTIONS
Name: ANSWE KEY Math 46: Probability MWF pm, Gasson Exam SOLUTIONS Problem Points Score 4 5 6 BONUS Total 6 Please write neatly. You may leave answers below unsimplified. Have fun and write your name above!
2 Functions of random variables
2 Functions of random variables A basic statistical model for sample data is a collection of random variables X 1,..., X n. The data are summarised in terms of certain sample statistics, calculated as
ECE Homework Set 2
1 Solve these problems after Lecture #4: Homework Set 2 1. Two dice are tossed; let X be the sum of the numbers appearing. a. Graph the CDF, FX(x), and the pdf, fx(x). b. Use the CDF to find: Pr(7 X 9).
Chapter 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
Continuous 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
On general error distributions
ProbStat Forum, Volume 6, October 3, Pages 89 95 ISSN 974-335 ProbStat Forum is an e-journal. For details please isit www.probstat.org.in On general error distributions R. Vasudea, J. Vasantha Kumari Department
Chapter 2: Random Variables (Cont d)
Civil Engineering Department: Engineering Statistics (ECIV 005) Chapter : Random Variables (Cont d) Section.6: Combinations and Functions of Random Variables Problem (): Suppose that the random variables
7 Random samples and sampling distributions
7 Random samples and sampling distributions 7.1 Introduction - random samples We will use the term experiment in a very general way to refer to some process, procedure or natural phenomena that produces
STAT 515 MIDTERM 2 EXAM November 14, 2018
STAT 55 MIDTERM 2 EXAM November 4, 28 NAME: Section Number: Instructor: In problems that require reasoning, algebraic calculation, or the use of your graphing calculator, it is not sufficient just to write
Homework 5 Solutions
126/DCP126 Probability, Fall 214 Instructor: Prof. Wen-Guey Tzeng Homework 5 Solutions 5-Jan-215 1. Let the joint probability mass function of discrete random variables X and Y be given by { c(x + y) ifx
ASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA Errata
ASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA (krzysio@krzysio.net) Errata Effective July 5, 3, only the latest edition of this manual will have its errata
Expectation, variance and moments
Expectation, variance and moments John Appleby Contents Expectation and variance Examples 3 Moments and the moment generating function 4 4 Examples of moment generating functions 5 5 Concluding remarks
Math 151. Rumbos Fall Solutions to Review Problems for Final Exam
Math 5. Rumbos Fall 23 Solutions to Review Problems for Final Exam. Three cards are in a bag. One card is red on both sides. Another card is white on both sides. The third card in red on one side and white
Summary. Ancillary Statistics What is an ancillary statistic for θ? .2 Can an ancillary statistic be a sufficient statistic?
Biostatistics 62 - Statistical Inference Lecture 5 Hyun Min Kang 1 What is an ancillary statistic for θ? 2 Can an ancillary statistic be a sufficient statistic? 3 What are the location parameter and the
HW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100]
![HW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100]](/thumbs/76/73074579.jpg "HW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100]")
HW7 Solutions. 5 pts.) James Bond James Bond, my favorite hero, has again jumped off a plane. The plane is traveling from from base A to base B, distance km apart. Now suppose the plane takes off from
Continuous 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
Review 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,
STA 4321/5325 Solution to Extra Homework 1 February 8, 2017
STA 431/535 Solution to Etra Homework 1 February 8, 017 1. Show that for any RV X, V (X 0. (You can assume X to be discrete, but this result holds in general. Hence or otherwise show that E(X E (X. Solution.
Sampling Distributions
In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of random sample. For example,
Transformations and Expectations
Transformations and Expectations 1 Distributions of Functions of a Random Variable If is a random variable with cdf F (x), then any function of, say g(), is also a random variable. Sine Y = g() is a function
EXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS
EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker
Course: ESO-209 Home Work: 1 Instructor: Debasis Kundu
Home Work: 1 1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times, (d) an infinite number of times. 2. A coin is tossed until for the first time the same result appear
Uses of Asymptotic Distributions: In order to get distribution theory, we need to norm the random variable; we usually look at n 1=2 ( X n ).
1 Economics 620, Lecture 8a: Asymptotics II Uses of Asymptotic Distributions: Suppose X n! 0 in probability. (What can be said about the distribution of X n?) In order to get distribution theory, we need
Chapter 5. Continuous Probability Distributions
Chapter 5. Continuous Probability Distributions Sections 5.2, 5.3: Expected Value of Continuous Random Variables and Uniform Distribution Jiaping Wang Department of Mathematical Science 03/20/2013, Monday
Chapter 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,
Limiting Distributions
Limiting Distributions We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the
Lecture 1: August 28
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 1: August 28 Our broad goal for the first few lectures is to try to understand the behaviour of sums of independent random
f X, Y (x, y)dx (x), where f(x,y) is the joint pdf of X and Y. (x) dx
INDEPENDENCE, COVARIANCE AND CORRELATION Independence: Intuitive idea of "Y is independent of X": The distribution of Y doesn't depend on the value of X. In terms of the conditional pdf's: "f(y x doesn't
Chapter 5. Continuous Probability Distributions
Chapter 5. Continuous Probability Distributions Sections 5.4, 5.5: Exponential and Gamma Distributions Jiaping Wang Department of Mathematical Science 03/25/2013, Monday Outline Exponential: PDF and CDF
1. Point Estimators, Review
AMS571 Prof. Wei Zhu 1. Point Estimators, Review Example 1. Let be a random sample from. Please find a good point estimator for Solutions. There are the typical estimators for and. Both are unbiased estimators.
2. Conditional Expectation (9/15/12; cf. Ross)
2. Conditional Expectation (9/15/12; cf. Ross) Intro / Definition Examples Conditional Expectation Computing Probabilities by Conditioning 1 Intro / Definition Recall conditional probability: Pr(A B) Pr(A
E X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.
E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,
SOLUTION FOR HOMEWORK 12, STAT 4351
SOLUTION FOR HOMEWORK 2, STAT 435 Welcome to your 2th homework. It looks like this is the last one! As usual, try to find mistakes and get extra points! Now let us look at your problems.. Problem 7.22.
3 Conditional Expectation
3 Conditional Expectation 3.1 The Discrete case Recall that for any two events E and F, the conditional probability of E given F is defined, whenever P (F ) > 0, by P (E F ) P (E)P (F ). P (F ) Example.
Math 341: Probability Seventeenth Lecture (11/10/09)
Math 341: Probability Seventeenth Lecture (11/10/09) Steven J Miller Williams College Steven.J.Miller@williams.edu http://www.williams.edu/go/math/sjmiller/ public html/341/ Bronfman Science Center Williams
Basic concepts of probability theory
Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential,
SOLUTIONS TO MATH68181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH68181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question A1 a) The marginal cdfs of F X,Y (x, y) = [1 + exp( x) + exp( y) + (1 α) exp( x y)] 1 are F X (x) = F X,Y (x, ) = [1
ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.
ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n
Math Real Analysis II
Math 432 - Real Analysis II Solutions to Homework due February 3 In class, we learned that the n-th remainder for a smooth function f(x) defined on some open interval containing is given by f (k) () R
5.2 Continuous random variables
5.2 Continuous random variables It is often convenient to think of a random variable as having a whole (continuous) interval for its set of possible values. The devices used to describe continuous probability
Recitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
Continuous distributions
CHAPTER 7 Continuous distributions 7.. Introduction A r.v. X is said to have a continuous distribution if there exists a nonnegative function f such that P(a X b) = ˆ b a f(x)dx for every a and b. distribution.)
Probability and Statistics Notes
Probability and Statistics Notes Chapter Five Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Five Notes Spring 2011 1 / 37 Outline 1
MATH Solutions to Probability Exercises
MATH 5 9 MATH 5 9 Problem. Suppose we flip a fair coin once and observe either T for tails or H for heads. Let X denote the random variable that equals when we observe tails and equals when we observe
BMIR 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
1 Dimension Reduction in Euclidean Space
CSIS0351/8601: Randomized Algorithms Lecture 6: Johnson-Lindenstrauss Lemma: Dimension Reduction Lecturer: Hubert Chan Date: 10 Oct 011 hese lecture notes are supplementary materials for the lectures.
MATH 104: INTRODUCTORY ANALYSIS SPRING 2008/09 PROBLEM SET 8 SOLUTIONS
MATH 04: INTRODUCTORY ANALYSIS SPRING 008/09 PROBLEM SET 8 SOLUTIONS. Let f : R R be continuous periodic with period, i.e. f(x + ) = f(x) for all x R. Prove the following: (a) f is bounded above below