The story of the film so far... Mathematics for Informatics 4a. Jointly distributed continuous random variables. José Figueroa-O Farrill
|
|
- George Randall
- 5 years ago
- Views:
Transcription
1 The story of the film so far... Mathematics for Informatics 4a José Figueroa-O Farrill Lecture 2 2 March 22 X a c.r.v. with p.d.f. f and g : R R: then Y g(x is a random variable and E(Y g(f(d variance: Var(X E(X 2 E(X 2 moment generating function: M X (t E(e tx have met uniform, eponential and normal distributions and have computed their mean, variance and m.g.f. if X normally distributed with mean µ and variance σ 2, Y σ (X µ has standard normal distribution The c.d.f. Φ of the standard normal distribution is not an elementary function, but there are tables maimum entropy: normal distribution is the least biased among all p.d.f.s with the same mean and variance José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2 Jointly distributed continuous random variables Two continuous random variables X and Y are said to be jointly distributed with joint density f(, y if for all a < b and c < d, P(a < X < b, c < Y < d d b c a f(, yddy It follows that the joint density obeys f(, y and Eample Let X and Y have joint density f(, y cy, y. What is c? From the normalisation condition, ( ( cy d dy c y2 c 4 c 4 and (provided C is nice enough f(, yddy P((X, Y C f(, yddy C What if < y? Since the density is symmetric in y, the integral over half the square is half of the previous result, hence c is twice the previous value: c 8. José Figueroa-O Farrill mi4a (Probability Lecture 2 3 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 4 / 2
2 Uniform joint densities Let A R 2 be a region with area A. X and Y are (jointly uniform in A if f(, y { A, (, y A, elsewhere Marginals Let X, Y be continuous random variables with joint density f(, y. Then the marginal p.d.f.s f X ( and f Y (y are given by f X ( f(, ydy and f Y (y f(, yd Eample Let X, Y be jointly uniform in the unit disk D {(, y 2 + y 2 }. Then D π, whence f(, y π 2 + y 2 Remark As in the discrete case there is no need to stop at two random variables, and we can have joint densities f(,..., n for n jointly distributed random variables, with many different marginals. José Figueroa-O Farrill mi4a (Probability Lecture 2 5 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 6 / 2 Eample Let X, Y be jointly uniform on the unit disk D: f(, y 2 + y 2 π Joint distributions Let X and Y be continuous random variables with joint density f(, y. Their joint distribution is defined as The marginals are given by F(, y P(X, Y y y f(u, vdu dv y 2 2 f X ( 2 π dy 2 2 π It follows from the fundamental theorem of calculus that 2 for and, by symmetry, f Y (y 2 π y 2 f(, y 2 F(, y y and the marginal distributions are obtained by for y F X ( F(, and F Y (y F(, y José Figueroa-O Farrill mi4a (Probability Lecture 2 7 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 8 / 2
3 Eample Let X, Y be jointly distributed with f(, y + y on, y. One checks that indeed ( + yddy. The joint distribution is F(, y y (u + vdu dv ( y (u + vdv du ( uy + 2 y2 du 2 2 y + 2 y2 for, y For y >, F(, y 2( + and similarly, for >, F(, y 2 y(y +. Independence Two continuous random variables X and Y are independent if or, equivalently, F(, y F X (F Y (y f(, y f X (f Y (y. It follows that for X, Y independent P(X A, Y B P(X AP(Y B Useful criterion: X and Y are independent iff f(, y g(h(y. Then f X ( cg( and f Y (y c h(y, where c R h(ydy. José Figueroa-O Farrill mi4a (Probability Lecture 2 9 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 Eamples X and Y are jointly uniform on a and y b: f(, y ab for (, y [, a] [, b] with marginals f X ( a and f Y(y b. Since f(, y f X (f Y (y, X and Y are independent. 2 X and Y are jointly uniform on the disk 2 + y 2 a 2 : f(, y πa 2 for 2 + y 2 a 2 with marginals f X ( πa a 2 2 and 2 f Y (y πa a 2 y 2. Since f(, y f 2 X (f Y (y, X and Y are not independent. Geometric probability Geometric probability or continuous combinatorics studies geometric objects sharing a common probability space. We have already seen some geometric probability problems in the tutorial sheets. For eample, in Tutorial Sheet 4 you considered the problem of tossing a coin on a square grid and computing the probability that the coin is fully contained inside one of the squares. This game was called franc-carreau ( free tile in France and was studied by Buffon in his treatise Sur le jeu de franc-carreau (733. In probability, Buffon is perhaps better known for Buffon s needle, which is a paradigmatic geometric probability problem. José Figueroa-O Farrill mi4a (Probability Lecture 2 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2
4 Buffon s needle I Drop a needle of length l at random on a striped floor, with stripes a distance L apart. Let l < L: short needles. L l What is the probability that the needle does not touch any line? Buffon s needle II The needle is described by the midpoint and the angle with the horizontal. Symmetry allows us to ignore the vertical component of the midpoint and to assume the horizontal component lies in one of the strips. Let X denote the horizontal component of the midpoint. It is uniformly distributed in [ L 2, L 2 ]. Let Θ denote the angle with the horizontal, which is uniformly distributed in [ π 2, π 2 ]. Since X and Θ are independent, the joint probability density function is the product of the two probability density functions and hence is also uniformly distributed. θ José Figueroa-O Farrill mi4a (Probability Lecture 2 3 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 4 / 2 Buffon s needle III The needle will touch one of the parallel lines if and only if + l 2 cos θ > L 2 for [ L 2, L 2 ] and θ [ π 2, π 2 ]. The complementary probability is P ( X 2 (L l cos Θ [ π 2 2 (L l cos θ π 2 π 2 π 2 π 2 l 2 (L l cos θ d (L l cos θdθ π 2 ] θ dθ cos θdθ 2l Functions of several random variables X and Y are continuous random variables with joint density f(, y Z g(x, Y, for some function g : R 2 R How is Z distributed? (assuming it is a c.r.v. Its c.d.f. F Z (z P(Z z is given by F Z (z f(, yd dy Its p.d.f. F Z (z g(,yz José Figueroa-O Farrill mi4a (Probability Lecture 2 5 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 6 / 2
5 Eample (The sum of two jointly uniform variables Let X and Y be jointly uniform on [, ] f(, y f X ( f Y (y for, y, so X and Y are independent. Let Z X + Y. F Z (z +yz d dy { 2 z 2, z [, ] 2 (2 z2, z [, 2] z y 2 z The sum of two independent variables: convolution Let X and Y be continuous random variables with joint density f(, y and let Z X + Y. Then F Z (z +yz f(, yd dy is given by ( z F Z (z f(, ydy d Hence F Z (z is given by f(, z d { z, z [, ] 2 z, z [, 2] 2 z If X and Y be independent, f(, y f X (f Y (y, whence f X (f Y (z d (f X f Y (z which defines the convolution product José Figueroa-O Farrill mi4a (Probability Lecture 2 7 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 8 / 2 Convolution The convolution product satisfies a number of interesting properties: commutativity: f g g f associativity: (f g h f (g h smoothing: f g is a smoother function than f or g, e.g., Summary C.r.v.s X and Y have a joint density f(, y with P((X, Y C f(, yd dy and a joint distribution C F(, y P(X, Y y y with f(, y 2 yf(, y X and Y independent iff f(, y f X (f Y (y Geometric probability is fun! (Buffon s needle f(u, vdu dv We can calculate the c.d.f. and p.d.f. of Z g(x, Y X, Y independent: f X+Y f X f Y (convolution José Figueroa-O Farrill mi4a (Probability Lecture 2 9 / 2 José Figueroa-O Farrill mi4a (Probability Lecture 2 2 / 2
Chapter 2. Random Variable. Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance.
Chapter 2 Random Variable CLO2 Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance. 1 1. Introduction In Chapter 1, we introduced the concept
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A real-valued
More information1 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
More informationLecture Notes 3 Multiple Random Variables. Joint, Marginal, and Conditional pmfs. Bayes Rule and Independence for pmfs
Lecture Notes 3 Multiple Random Variables Joint, Marginal, and Conditional pmfs Bayes Rule and Independence for pmfs Joint, Marginal, and Conditional pdfs Bayes Rule and Independence for pdfs Functions
More informationChapter 4 Multiple Random Variables
Chapter 4 Multiple Random Variables Chapter 41 Joint and Marginal Distributions Definition 411: An n -dimensional random vector is a function from a sample space S into Euclidean space n R, n -dimensional
More informationORF 245 Fundamentals of Statistics Joint Distributions
ORF 245 Fundamentals of Statistics Joint Distributions Robert Vanderbei Fall 2015 Slides last edited on November 11, 2015 http://www.princeton.edu/ rvdb Introduction Joint Cumulative Distribution Function
More informationConditional distributions (discrete case)
Conditional distributions (discrete case) The basic idea behind conditional distributions is simple: Suppose (XY) is a jointly-distributed random vector with a discrete joint distribution. Then we can
More informationChapter 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
More informationP (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
More informationPCMI Introduction to Random Matrix Theory Handout # REVIEW OF PROBABILITY THEORY. Chapter 1 - Events and Their Probabilities
PCMI 207 - Introduction to Random Matrix Theory Handout #2 06.27.207 REVIEW OF PROBABILITY THEORY Chapter - Events and Their Probabilities.. Events as Sets Definition (σ-field). A collection F of subsets
More informationMore on Distribution Function
More on Distribution Function The distribution of a random variable X can be determined directly from its cumulative distribution function F X. Theorem: Let X be any random variable, with cumulative distribution
More informationProblem Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2},
ECE32 Spring 25 HW Solutions April 6, 25 Solutions to HW Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 18
EECS 7 Discrete Mathematics and Probability Theory Spring 214 Anant Sahai Note 18 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces Ω,
More informationNotes 12 Autumn 2005
MAS 08 Probability I Notes Autumn 005 Conditional random variables Remember that the conditional probability of event A given event B is P(A B) P(A B)/P(B). Suppose that X is a discrete random variable.
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 18-27 Review Scott Sheffield MIT Outline Outline It s the coins, stupid Much of what we have done in this course can be motivated by the i.i.d. sequence X i where each X i is
More informationMultivariate distributions
CHAPTER Multivariate distributions.. Introduction We want to discuss collections of random variables (X, X,..., X n ), which are known as random vectors. In the discrete case, we can define the density
More informationCh. 5 Joint Probability Distributions and Random Samples
Ch. 5 Joint Probability Distributions and Random Samples 5. 1 Jointly Distributed Random Variables In chapters 3 and 4, we learned about probability distributions for a single random variable. However,
More informationE 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,
More informationsheng@mail.ncyu.edu.tw Content Joint distribution functions Independent random variables Sums of independent random variables Conditional distributions: discrete case Conditional distributions: continuous
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 17: Continuous random variables: conditional PDF Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 8 10/1/2008 CONTINUOUS RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 8 10/1/2008 CONTINUOUS RANDOM VARIABLES Contents 1. Continuous random variables 2. Examples 3. Expected values 4. Joint distributions
More informationECON 5350 Class Notes Review of Probability and Distribution Theory
ECON 535 Class Notes Review of Probability and Distribution Theory 1 Random Variables Definition. Let c represent an element of the sample space C of a random eperiment, c C. A random variable is a one-to-one
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationMath 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.
Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample
More informationPerhaps the simplest way of modeling two (discrete) random variables is by means of a joint PMF, defined as follows.
Chapter 5 Two Random Variables In a practical engineering problem, there is almost always causal relationship between different events. Some relationships are determined by physical laws, e.g., voltage
More informationENGG2430A-Homework 2
ENGG3A-Homework Due on Feb 9th,. Independence vs correlation a For each of the following cases, compute the marginal pmfs from the joint pmfs. Explain whether the random variables X and Y are independent,
More informationStochastic processes Lecture 1: Multiple Random Variables Ch. 5
Stochastic processes Lecture : Multiple Random Variables Ch. 5 Dr. Ir. Richard C. Hendriks 26/04/8 Delft University of Technology Challenge the future Organization Plenary Lectures Book: R.D. Yates and
More information6.041/6.431 Fall 2010 Quiz 2 Solutions
6.04/6.43: Probabilistic Systems Analysis (Fall 200) 6.04/6.43 Fall 200 Quiz 2 Solutions Problem. (80 points) In this problem: (i) X is a (continuous) uniform random variable on [0, 4]. (ii) Y is an exponential
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 information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 16. A Brief Introduction to Continuous Probability
CS 7 Discrete Mathematics and Probability Theory Fall 213 Vazirani Note 16 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces Ω, where the
More informationMath 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
More informationUNIT Define joint distribution and joint probability density function for the two random variables X and Y.
UNIT 4 1. Define joint distribution and joint probability density function for the two random variables X and Y. Let and represent the probability distribution functions of two random variables X and Y
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationLecture Notes 2 Random Variables. Discrete Random Variables: Probability mass function (pmf)
Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution
More information2 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
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 informationCS145: Probability & Computing
CS45: Probability & Computing Lecture 0: Continuous Bayes Rule, Joint and Marginal Probability Densities Instructor: Eli Upfal Brown University Computer Science Figure credits: Bertsekas & Tsitsiklis,
More informationLecture 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
More informationECE302 Exam 2 Version A April 21, You must show ALL of your work for full credit. Please leave fractions as fractions, but simplify them, etc.
ECE32 Exam 2 Version A April 21, 214 1 Name: Solution Score: /1 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationProbability, Random Processes and Inference
INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION Laboratorio de Ciberseguridad Probability, Random Processes and Inference Dr. Ponciano Jorge Escamilla Ambrosio pescamilla@cic.ipn.mx
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationEstimation theory. Parametric estimation. Properties of estimators. Minimum variance estimator. Cramer-Rao bound. Maximum likelihood estimators
Estimation theory Parametric estimation Properties of estimators Minimum variance estimator Cramer-Rao bound Maximum likelihood estimators Confidence intervals Bayesian estimation 1 Random Variables Let
More informationProbability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
More informationRandom Variables and Probability Distributions
CHAPTER Random Variables and Probability Distributions Random Variables Suppose that to each point of a sample space we assign a number. We then have a function defined on the sample space. This function
More informationRandom Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline.
Random Variables Amappingthattransformstheeventstotherealline. Example 1. Toss a fair coin. Define a random variable X where X is 1 if head appears and X is if tail appears. P (X =)=1/2 P (X =1)=1/2 Example
More informationBMIR 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
More informationTwo hours. Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER. 14 January :45 11:45
Two hours Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER PROBABILITY 2 14 January 2015 09:45 11:45 Answer ALL four questions in Section A (40 marks in total) and TWO of the THREE questions
More informationLecture Notes 2 Random Variables. Random Variable
Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution
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 informationMATH/STAT 395 Winter 2013 This homework is due at the beginning of class on Friday, March 1.
Homework 6 KEY MATH/STAT 395 Winter 23 This homework is due at the beginning of class on Friday, March.. A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. The net
More informationContents 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................
More informationMath/Stats 425, Sec. 1, Fall 04: Introduction to Probability. Final Exam: Solutions
Math/Stats 45, Sec., Fall 4: Introduction to Probability Final Exam: Solutions. In a game, a contestant is shown two identical envelopes containing money. The contestant does not know how much money is
More informationSOLUTION FOR HOMEWORK 11, ACTS 4306
SOLUTION FOR HOMEWORK, ACTS 36 Welcome to your th homework. This is a collection of transformation, Central Limit Theorem (CLT), and other topics.. Solution: By definition of Z, Var(Z) = Var(3X Y.5). We
More informationRandom 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
More information18.440: Lecture 19 Normal random variables
18.440 Lecture 19 18.440: Lecture 19 Normal random variables Scott Sheffield MIT Outline Tossing coins Normal random variables Special case of central limit theorem Outline Tossing coins Normal random
More informationSTAT 414: Introduction to Probability Theory
STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises
More informationHW4 : Bivariate Distributions (1) Solutions
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 7 Néhémy Lim HW4 : Bivariate Distributions () Solutions Problem. The joint probability mass function of X and Y is given by the following table : X Y
More informationSTAT 430/510: Lecture 16
STAT 430/510: Lecture 16 James Piette June 24, 2010 Updates HW4 is up on my website. It is due next Mon. (June 28th). Starting today back at section 6.7 and will begin Ch. 7. Joint Distribution of Functions
More informationChapter 5. Random Variables (Continuous Case) 5.1 Basic definitions
Chapter 5 andom Variables (Continuous Case) So far, we have purposely limited our consideration to random variables whose ranges are countable, or discrete. The reason for that is that distributions on
More informationThe story of the film so far... Mathematics for Informatics 4a. Continuous-time Markov processes. Counting processes
The story of the film so far... Mathematics for Informatics 4a José Figueroa-O Farrill Lecture 19 28 March 2012 We have been studying stochastic processes; i.e., systems whose time evolution has an element
More informationAppendix A : Introduction to Probability and stochastic processes
A-1 Mathematical methods in communication July 5th, 2009 Appendix A : Introduction to Probability and stochastic processes Lecturer: Haim Permuter Scribe: Shai Shapira and Uri Livnat The probability of
More informationIIT JAM : MATHEMATICAL STATISTICS (MS) 2013
IIT JAM : MATHEMATICAL STATISTICS (MS 2013 Question Paper with Answer Keys Ctanujit Classes Of Mathematics, Statistics & Economics Visit our website for more: www.ctanujit.in IMPORTANT NOTE FOR CANDIDATES
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 17-27 Review Scott Sheffield MIT 1 Outline Continuous random variables Problems motivated by coin tossing Random variable properties 2 Outline Continuous random variables Problems
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 informationSolution 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
More informationJoint Distribution of Two or More Random Variables
Joint Distribution of Two or More Random Variables Sometimes more than one measurement in the form of random variable is taken on each member of the sample space. In cases like this there will be a few
More informationNotes for Math 324, Part 20
7 Notes for Math 34, Part Chapter Conditional epectations, variances, etc.. Conditional probability Given two events, the conditional probability of A given B is defined by P[A B] = P[A B]. P[B] P[A B]
More informationECE353: 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
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 information1 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
More informationLecture The Sample Mean and the Sample Variance Under Assumption of Normality
Math 408 - Mathematical Statistics Lecture 13-14. The Sample Mean and the Sample Variance Under Assumption of Normality February 20, 2013 Konstantin Zuev (USC) Math 408, Lecture 13-14 February 20, 2013
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationADVANCED PROBABILITY: SOLUTIONS TO SHEET 1
ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Last compiled: November 6, 213 1. Conditional expectation Exercise 1.1. To start with, note that P(X Y = P( c R : X > c, Y c or X c, Y > c = P( c Q : X > c, Y
More informationDiscrete Mathematics and Probability Theory Fall 2015 Note 20. A Brief Introduction to Continuous Probability
CS 7 Discrete Mathematics and Probability Theory Fall 215 Note 2 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces Ω, where the number
More informationMathematics 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
More informationconditional cdf, conditional pdf, total probability theorem?
6 Multiple Random Variables 6.0 INTRODUCTION scalar vs. random variable cdf, pdf transformation of a random variable conditional cdf, conditional pdf, total probability theorem expectation of a random
More information3. Probability and Statistics
FE661 - Statistical Methods for Financial Engineering 3. Probability and Statistics Jitkomut Songsiri definitions, probability measures conditional expectations correlation and covariance some important
More informationHW1 (due 10/6/05): (from textbook) 1.2.3, 1.2.9, , , (extra credit) A fashionable country club has 100 members, 30 of whom are
HW1 (due 10/6/05): (from textbook) 1.2.3, 1.2.9, 1.2.11, 1.2.12, 1.2.16 (extra credit) A fashionable country club has 100 members, 30 of whom are lawyers. Rumor has it that 25 of the club members are liars
More informationMATH 151, FINAL EXAM Winter Quarter, 21 March, 2014
Time: 3 hours, 8:3-11:3 Instructions: MATH 151, FINAL EXAM Winter Quarter, 21 March, 214 (1) Write your name in blue-book provided and sign that you agree to abide by the honor code. (2) The exam consists
More informationACM 116: Lectures 3 4
1 ACM 116: Lectures 3 4 Joint distributions The multivariate normal distribution Conditional distributions Independent random variables Conditional distributions and Monte Carlo: Rejection sampling Variance
More informationECE 650 Lecture 4. Intro to Estimation Theory Random Vectors. ECE 650 D. Van Alphen 1
EE 650 Lecture 4 Intro to Estimation Theory Random Vectors EE 650 D. Van Alphen 1 Lecture Overview: Random Variables & Estimation Theory Functions of RV s (5.9) Introduction to Estimation Theory MMSE Estimation
More information2 Statistical Estimation: Basic Concepts
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof. N. Shimkin 2 Statistical Estimation:
More informationChapter 5,6 Multiple RandomVariables
Chapter 5,6 Multiple RandomVariables ENCS66 - Probabilityand Stochastic Processes Concordia University Vector RandomVariables A vector r.v. is a function where is the sample space of a random experiment.
More informationRandom variable X is a mapping that maps each outcome s in the sample space to a unique real number x, < x <. ( ) X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number, <
More informationCS145: 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
More informationReview: mostly probability and some statistics
Review: mostly probability and some statistics C2 1 Content robability (should know already) Axioms and properties Conditional probability and independence Law of Total probability and Bayes theorem Random
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4. (*. Let independent variables X,..., X n have U(0, distribution. Show that for every x (0,, we have P ( X ( < x and P ( X (n > x as n. Ex. 4.2 (**. By using induction or otherwise,
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 informationSTAT 418: Probability and Stochastic Processes
STAT 418: Probability and Stochastic Processes Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical
More informationfor valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I
Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks:
More informationTHE QUEEN S UNIVERSITY OF BELFAST
THE QUEEN S UNIVERSITY OF BELFAST 0SOR20 Level 2 Examination Statistics and Operational Research 20 Probability and Distribution Theory Wednesday 4 August 2002 2.30 pm 5.30 pm Examiners { Professor R M
More informationPart IA Probability. Theorems. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Theorems Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationNonparametric hypothesis tests and permutation tests
Nonparametric hypothesis tests and permutation tests 1.7 & 2.3. Probability Generating Functions 3.8.3. Wilcoxon Signed Rank Test 3.8.2. Mann-Whitney Test Prof. Tesler Math 283 Fall 2018 Prof. Tesler Wilcoxon
More informationSTA 4321/5325 Solution to Homework 5 March 3, 2017
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 4 + 4 8. Therefore,
More informationn px p x (1 p) n x. p x n(n 1)... (n x + 1) x!
Lectures 3-4 jacques@ucsd.edu 7. Classical discrete distributions D. The Poisson Distribution. If a coin with heads probability p is flipped independently n times, then the number of heads is Bin(n, p)
More informationContinuous 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.)
More information