Penn State, STAT 401

Excerpt: ... Outline The Joint Probability Density Function Statistics and Sampling Distributions Lecture 14 Chapter 4: Multivariate Variables and Their Distribution Michael Akritas Michael Akritas Lecture 14 Chapter 4: Multivariate Variables and Their Distribu Outline The Joint Probability Density Function Statistics and Sampling Distributions The Joint Probability Density Function Denition and Basic Properties Marginal Probability Density Functions Conditional Probability Density Functions Independence Statistics and Sampling Distributions Denitions and Examples Michael Akritas Lecture 14 Chapter 4: Multivariate Variables and Their Distribu Outline The Joint Probability Density Function Statistics and Sampling Distributions Denition and Basic Properties Marginal Probability Density Functions Conditional Probability Density Functions Independence The pdf of a univariate r.v. X is a function f (x) such that probabilities are represented as areas under the curve. Michael Akritas Lecture 14 Chapter 4 ...

UIllinois, STAT 400

Excerpt: ... m variables X and Y are independent if P(X=x,Y=y)=P(X=x)P(Y=y) Or, equivalently, f(x,y)=f1(x)f2(y) Example: f(x,y)=(x+y)/21 for x=1,2,3 and y=1,2. (a)Find the marginal pmf of X (b) Are X and Y independent? Ping Ma Lecture 18 Fall 2005 -2- Mathematical Expectation Definition: If f(x,y) is the p.m.f. of discrete random variables X andY. Then the mathematical expectation or the expected value of function u(X,Y) E[u(X,Y)]= u ( x, y ) f ( x, y ) Example: Let pmf and X1 and X2 are f(x1,x2)=(3-x1-x2)/8 for x1=0,1 and x2=0,1. Find E[X+Y] Ping Ma Lecture 18 Fall 2005 -3- Example: DNA sequence consists of four nucleotides A, T C and G. A CpG island, a special region of a DNA sequence, has the 40% C, 40% G, and 20% A and T. If we choose 15 nucleotides randomly for a study, what is the probability that we have 6 Cs and 7 Gs? Continuous random variables Joint probability density function Properties of joint p.d.f f(x,y): (a) f(x,y)>0; (b) f ( x, y)dxdy 1 ; (c) P(a<X<b, c<Y<d)= b a ...

Michigan, STAT 425

Excerpt: ... Stat 425 Problems due 4/11/06 Chapter 7: 9a (there can be more than one ball in an urn), 11, 32, 34 A) An environmental engineer measures the amount (by weight) of particulate pollution in air samples of a certain volume collected over the smokestack of a coal-operated power plant. Let Y1 denote the amount of pollutant per sample collected when the cleaning device on the stack is not operating and let Y2 denote the amount of pollutant per sample collected under the same environmental conditions when the cleaning device is operating. It is observed that the joint probability density function of Y1 and Y2 can be modeled by fY1 Y2 (y1 , y2 ) = K 0 if 0 y1 2; 0 y2 1; 2y2 y1 ; elsewhere. (That is, Y1 and Y2 are uniformly distributed over the region inside the triangle bounded by y1 = 2, y2 = 0 and 2y2 = y1 .) 1. Find the value of K that makes this function a joint probability density function . 2. Find P (Y1 3Y2 ). (That is, find the probability that the cleaning device will reduce the amount of pollutant ...

UCSC, AMS 131

Excerpt: ... Name: Discussion Section: AMS-131: Introduction to Probability Theory Quiz 4 (Thursday May 28, 2009) Quizzes are closed-book, closed-notes, with the exception of one (letter size) piece of paper with formulas on both sides. Please show all your work. Unsupported answers will receive little (or no) credit, even if they are correct. Good luck! Suppose that X and Y have a continuous joint distribution for which the joint probability density function is given by f (x, y) = 3 2 -y 8x e 0 for 0 x 2 and y > 0 otherwise (a) (4 points) Compute the probability Pr(X > 1 and Y < 1). (b) (6 points) Obtain the marginal probability density functions f 1 (x) and f2 (y) for X and Y , respectively. Are X and Y independent? Justify your answer. ...

UConn, STAT 230

Excerpt: ... probability mass function of Y1 is: 2 3 y1 6 2 2 y2 =0 p(y1 ) = y2 =0 3 y1 p(y1 , y2 ) = 3 2-y1 6 2 2 y2 1 2 - y1 - y2 = y1 = 0, 1, 2 Similarly, The marginal probability mass function of Y2 is 2 p(y2 ) = y1 =0 2 y2 p(y1 , y2 ) = 4 2-y2 6 2 2 y2 6 2 2 y1 =0 3 y1 1 2 - y1 - y2 = y2 = 0, 1, 2 Multivariate Probability Distributions: Part I p. 12/3 Jointly Distributed Discrete Random Variables All these information can be summarized in the table below. y1 0 0 y2 1 2 p1 (y1 ) 0 2/15 1/15 3/15 1 3/15 6/15 0 9/15 2 3/15 0 0 3/15 p2 (y2 ) 6/15 8/15 1/15 1 Multivariate Probability Distributions: Part I p. 13/3 Jointly Distributed Continuous Random Variables Definition 5.4: Let Y1 and Y2 be two continuous random variables associated to a random experiment. The joint probability density function , f (y1 , y2 ) for Y1 and Y2 is the function such that for any two-dimensional set A P (Y1 , Y2 ) A = (y1 ,y2 )A f (y1 , y2 ) dy1 dy2 . In particular, b d P a Y1 b, c Y2 d = a c f (y1 , y2 ) dy ...

Wisconsin, STAT 312

Textbook: Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

Excerpt: ... Stat 312: Lecture 22 Inference on Correlation Moo K. Chung mchung@stat.wisc.edu April 17, 2003 Concepts 1. Assume that X and Y are two random variables whose joint probability density function f is the 2 bivariate normal such that X N (X , X ), Y 2 ) and (X, Y ) = . In this situation, we N (Y , Y can show that (Y |X = x) = 0 + 1 x, where 1 = Y /X . This is the regression model we studied! procedure and end up with the same result. Example. Test for the dependence of the mathematical achievement test scores (xi ) and calculus grades (yi ) for 10 college freshmen assuming (x i , yi ) are from bivariate normal. x 39, 43, 21, 64, 57, 47, 28, 75, 34, 52 y 65, 78, 52, 82, 92, 89, 73, 98, 56, 75 Solution. There are two ways to solve this problem. 2. Testing H0 : = 0 vs. H1 : = 0 is equiva- The first approach might be much simpler conceptually. lent to H0 : 1 = 0 vs. H1 : 1 = 0. So the The only thing you need is to compute the correlation appropriate test statistic for testing H 0 : = 0 is coeffi ...

Iowa State, STAT 341

Excerpt: ... Statistics 341, Exam 3, Friday, April 14, 2006, Michael Larsen Closed book. No notes. Calculator permitted. Read directions carefully. Show work. Answers without adequate work and explanation might receive reduced or no credit. DISTRIBUTION FORMULAS PROVIDED. NORMAL TABLE PROVIDED. NAME: Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Total Parts Points Score 3 20 2 13 3 20 2 14 2 13 4 20 16 100 f (x) = 12x(1 - x)2 1. (20 points) Let X have the following probability density function (pdf): for 0 < x < 1 and f (x) = 0 elsewhere. (a) Show that the function is, in fact, a probability density function (pdf). (b) What is the mean of X? (c) What is the probability that X has a value between -0.1 and 0.2? 2. (13 points) Consider the following joint probability density function : 2 f (x, y) = ye-x/3 3 for 0 < y < 1 and 0 < x < and f (x, y) = 0 otherwise. (a) Are X and Y independent? Why or why not? (b) What is E(XY )? Show work or briefly explain how you arrive at your answer. 3. (20 points) Let rando ...

SUNY Stony Brook, AMS 311

Excerpt: ... AMS 311 (Fall, 2008) Joe Mitchell PROBABILITY THEORY Homework Set # 7 Due at the beginning of class on Tuesday, November 25, 2008. Reminder: Show your reasoning! You do not need to evaluate arithmetic expressions or integrals, if they are fully specied. For example, 0.5 1x 2 y you may leave 0 x e dydx in this form. x Read: Ross, Chapter 6, Sections 6.4, 6.5; Chapter 7, Section 7.5 and Section 7.7; and handout Notes on Expectation, Moment Generating Functions, Variance, Covariance SPECIFICS OF READING ASSIGNMENT: Examples to read carefully: Chapter 6: 4a, 4b, 5a, 5b Chapter 7: 5a, 5b, 5c, 5d, 5k, 5l, 7a, 7b, 7d, 7e, 7f, 7g, 7h (1). (15 points) If E(3X) = var(X/2) and var(2X) = 3, nd (a). E[(2 + X)2 ] and (b). var(4 + 3X). (2). (15 points) The random variables X and Y have a joint density function given by f (x, y) = Compute cov(2X, Y + 3). (3). (30 points) Let X and Y be continuous random variables with joint probability density function given by 2 f (x, y) = Cx if x 0, x < 4, x y, ...

East Los Angeles College, EC 961

Excerpt: ... c Y d) = f ( x, y)dydx a c b d This function is called a joint probability density function . The joint probability function must also be nonnegative and sum to one. The cumulative joint distribution function is given by: F ( x, y ) = P( X x and Y y ) = P (- X x and - Y y ) = - - f (s, t )dtds x y Properties of Joint cdfs 1.F (- ,- ) = 0 2.F (, ) = 1 2 F ( x, y ) 3. = f ( x, y ) xy 4.P (a X b and c Y d ) = F (b, d ) - F (a, d ) - F (b, c) + F (a, c) Marginal Distributions and Independence To recover the marginal distribution in the continuous case, we simply integrate out the relevant variable. Thus: g ( x) = f ( x, y )dy - h( y ) = f ( x, y )dx - Recall that we said that two variables were independent if P(AB) = P(A)P(B). Similarly, two random variables are independent if f(x,y) = g(x)h(y). That is to say if we can write the joint probability density function as the product of the marginal distributions. Expectation and Variance I will only consider the continuous cas ...

Oakland University, MATH 30530

Excerpt: ... Math 30530-02, Fall 2007 Review for Exam 3 Exam 3, Tuesday, November 27 at 8 a.m. in DBRT 140, will cover sections 7.1-7.4 and 8.1-8.3. Of course this material builds on the earlier material. Section 9.1 wont be covered on this exam but is good reinforcement for 8.1-8.3. You are allowed to use an 8 1 11 sheet of paper with notes in your own handwriting on one side (or 2 an index card up to 5 8 with notes on both sides). A table of values of will be provided. Here is an outline of the topics. 1. Specic types of continuous distributions Uniform Normal Normal approximation to the binomial distribution Correction for continutity Exponential Gamma 2. Jointly distributed random variables Joint cumulative probability distribution function (joint distribution function) Marginal distribution functions Joint probability mass function Relation with probability mass functions Jointly continuous random variables Joint probability density function (joint densit ...