Continuous Distributions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Continuous Distributions"

Transcription

1 MAT X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take values an interval of real numbers. We will call such variables continuous. Examples of continuous random variables: length, area, volume, pressure, temperature, mass and many others. For a continuous random variable X will also specify its probability distribution in two different ways: 1. with its cumulative distribution function (c.d.f.) which is F X (x) = P (X x), where x is a real number; 2. with its probability density function. We will define the density below. We have seen that for discrete random variables that the cumulative distribution function is a non-decreasing function step function. As we encounter a value in the range of the random variable, then a probability mass is added to the cumulation of probabilities. Now if all values in an interval are possible values, then we could cumulate probabilities continuously. This motivates the following definition. 1

2 Definition: Let X be a random variable with cumulative distribution function F X. If the function F X is continuous, then we say the random variable X is continuous. Definition : Let X be a continuous random variable with c.d.f. F. We define its probability density function (p.d.f.) as a function f X such that { F f X (x) = X (x), if it exists, 0, otherwise, where F is the derivative function of F. In other words, the density is the rate at which the probabilities are cumulated. By the Fundamental Theorem of Calculus, we can obtain another interpretation of the density: P (a < X b) = P (X b) P (X < a) Remarks: = F X (b) F X (a) = b a f X (x) dx. So the probability that X will fall in the interval (a, b] is the area under the probability density from x = a to x = b. A single real number will not have a mass, that is for a continuous random variable X, P (X = x) = 0, for all x 2

3 For a, b R, such that a < b, then P (a < X < b) = P (a X < b) = P (a X b) = P (a < X b) = See the accompanying graph: b a f(x) dx = F (b) F (a) It is the graph of a cdf and the corresponding density. The area under the density from x = 39.1 to x = 53 is This means that P (39.1 < X < 53) = Interpretation: As we repeat the experiment a large number of times about 23% of the values should fall between 39.1 and 53. 3

4 Properties of a p.d.f. : 1. f(x) 0 2. f(x) dx = 1 3. [Computational Property] Let A R then P (X A) = f(x) dx In particular, let F be the c.d.f. for X, then A F (x) = P (X x) = x f(t) dt. Definition: Let X be a continuous random variable with probability density function f X. The expected value of X is defined as E[X] = x f(x) dx. Definition: Let X be a continuous random variable with probability density function f. Its mean is defined as µ X = E[X] = Its variance is defined as σ 2 X = Var(X) = E[(X µ) 2 ] = x f(x) dx. Its standard deviation is defined as σ X = Var(X). (x µ) 2 f(x) dx 4

5 Remark: The mean, the variance and the standard deviation for a continuous random variable are interpreted in the same way as the corresponding measures for a discrete random variable. The mean of X represents the center of mass of the distribution and also the expected value of the random variable. The variance of X is a measure of the variability or dispersion of the values about the mean, in units squared. The standard deviation also measures the variability or the dispersion of the values about the mean, but in the same units as the original measurements. 5

6 We now present a distribution, known as the normal distribution, that is often used as an approximation to the true distribution of a random variable. It is often a reasonable model but not always. We will see a theorem later that will explain why the normal distribution is often a reasonable model. Normal Distribution Definition: A continuous random variable X with p.d.f. f(x) = 1 2πσ 2 e (x µ)2 /(2 σ 2), < x < is said to follow a normal distribution with parameters µ and σ, where < µ < and σ > 0. Note: Let X be a normal random variable with parameters µ and σ, then its mean and variance are respectively E[X] = µ and Var[X] = σ 2. Notation: X N(µ, σ 2 ) will mean that X follows a normal distribution with mean µ and variance σ 2. 6

7 Properties of Normal Curves: 1. The graph of the density of any normal random variable is symmetric, bell-shaped curve centered about its mean µ. Note: We call µ a location parameter. 2. The points of inflection in the curve occur for values of X one standard deviation away from the mean, i.e. at the values x = µ ± σ. Note: We call σ a shape parameter. Empirical Rule: about 2/3 of the values are within 1 standard deviation from the mean; about 95% of the values are within 2 standard deviations from the mean; about 99.7% of the values are within 3 standard deviations from the mean. 7

8 Definition: A standard normal random variable Z is a normal random variable with mean E[Z] = 0 and variance Var(Z) = 1. Its p.d.f. is Its c.d.f is φ(z) = 1 2π e z2 /2, < z < Φ(z) = P (Z z) = = z z φ(x) dx 1 2π e x2 /2 dx Remark: Some values of Φ(z) = P [Z z] are found in a table given on the web page accompanying these notes. We will also learn to use R to compute these values. Properties of the standard normal : Let Z be a standard normal random variable, then 1. its p.d.f φ is symmetric about the origin, i.e. z = 0; 2. Φ( z) = 1 Φ(z), that is P (Z z) = P (Z z). 8

9 Example 1: Using the table for the standard normal and also using Minitab answer the following questions. Let Z be a standard normal random variable, that is Z follows a N(0, 1) distribution. Find 1. P (.53 < Z < 2.06) 2. P ( 2.63 Z.51) 3. P ( Z > 1.96) 4. c such that P ( c Z c) = c such that P (Z > c) = c such that P (Z < c) =.99 Standardization Theorem: If X is a normal random variable with mean E[X] and variance V [X], then Z = X E[X] σ X, ( where σ X = V [X]), is a standard normal random variable. Consequences of the standardization theorem: ( X E[X] P (X x) = P x E[X] ) σ X σ ( X = P Z x E[X] ) ( ) x E[X] = Φ. σ X σ X and ( ) ( ) d E[X] c E[X] P (c X d) = Φ Φ σ X σ X 9

10 Example 2: Assuming that among diabetics, the fasting blood glucose level X (in mg per 100 ml) may be assumed to be approximately normally distributed with mean 106 and standard deviation 8. (a) What percentage of diabetics have fasting blood glucose levels between 90 and 120? (b) Find a level x such that 25% of diabetics have a fasting glucose level lower than x. (c) If we selected 5 diabetics at random, what is the probability that at most 1 would have fasting blood glucose level between 90 and 120? 10

Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

More information

Continuous Probability Distributions (120 Points)

Continuous Probability Distributions (120 Points) Economics : Statistics for Economics Problem Set 5 - Suggested Solutions University of Notre Dame Instructor: Julio Garín Spring 22 Continuous Probability Distributions 2 Points. A management consulting

More information

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage 4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

More information

Example. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have

Example. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have Lectures 9-10 jacques@ucsd.edu 5.1 Random Variables Let (Ω, F, P ) be a probability space. The Borel sets in R are the sets in the smallest σ- field on R that contains all countable unions and complements

More information

10 BIVARIATE DISTRIBUTIONS

10 BIVARIATE DISTRIBUTIONS BIVARIATE DISTRIBUTIONS After some discussion of the Normal distribution, consideration is given to handling two continuous random variables. The Normal Distribution The probability density function f(x)

More information

Continuous random variables

Continuous random variables Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

More information

Lecture Note 7 Random Sample. MIT Spring 2006 Herman Bennett

Lecture Note 7 Random Sample. MIT Spring 2006 Herman Bennett Lecture Note 7 Random Sample MIT 14.30 Spring 2006 Herman Bennett 17 Definitions 17.1 Random Sample Let X 1,..., X n be mutually independent RVs such that f Xi (x) = f Xj (x) i = j. Denote f Xi (x) = f(x).

More information

Continuous Random Variables

Continuous Random Variables Continuous Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Continuous Random Variables 2 11 Introduction 2 12 Probability Density Functions 3 13 Transformations 5 2 Mean, Variance and Quantiles

More information

Discrete and Continuous Random Variables. Summer 2003

Discrete and Continuous Random Variables. Summer 2003 Discrete and Continuous Random Variables Summer 003 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random

More information

Statistiek (WISB361)

Statistiek (WISB361) Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

More information

Sampling Distribution of a Normal Variable

Sampling Distribution of a Normal Variable Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

More information

Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

More information

Jointly Distributed Random Variables

Jointly Distributed Random Variables Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

More information

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem 1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

More information

The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

More information

CSE 312, 2011 Winter, W.L.Ruzzo. 7. continuous random variables

CSE 312, 2011 Winter, W.L.Ruzzo. 7. continuous random variables CSE 312, 2011 Winter, W.L.Ruzzo 7. continuous random variables continuous random variables Discrete random variable: takes values in a finite or countable set, e.g. X {1,2,..., 6} with equal probability

More information

Chapter 4 Expected Values

Chapter 4 Expected Values Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

More information

Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

More information

Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty. Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

More information

Topic 2: Scalar random variables. Definition of random variables

Topic 2: Scalar random variables. Definition of random variables Topic 2: Scalar random variables Discrete and continuous random variables Probability distribution and densities (cdf, pmf, pdf) Important random variables Expectation, mean, variance, moments Markov and

More information

MATH 201. Final ANSWERS August 12, 2016

MATH 201. Final ANSWERS August 12, 2016 MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

More information

Random Variables. Chapter 2. Random Variables 1

Random Variables. Chapter 2. Random Variables 1 Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

More information

Summary of Probability

Summary of Probability Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

More information

Math/Stat 370: Engineering Statistics, Washington State University

Math/Stat 370: Engineering Statistics, Washington State University Math/Stat 370: Engineering Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering Statistics,

More information

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day.

Review Exam Suppose that number of cars that passes through a certain rural intersection is a Poisson process with an average rate of 3 per day. Review Exam 2 This is a sample of problems that would be good practice for the exam. This is by no means a guarantee that the problems on the exam will look identical to those on the exam but it should

More information

Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

More information

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

More information

Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

More information

POL 571: Expectation and Functions of Random Variables

POL 571: Expectation and Functions of Random Variables POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior

More information

Quantile-Quantile Plot (QQ-plot) and the Normal Probability Plot. Section 6-6 : Normal Probability Plot. MAT 2377 (Winter 2012)

Quantile-Quantile Plot (QQ-plot) and the Normal Probability Plot. Section 6-6 : Normal Probability Plot. MAT 2377 (Winter 2012) MAT 2377 (Winter 2012) Quantile-Quantile Plot (QQ-plot) and the Normal Probability Plot Section 6-6 : Normal Probability Plot Goal : To verify the underlying assumption of normality, we want to compare

More information

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

More information

Random Variables of The Discrete Type

Random Variables of The Discrete Type Random Variables of The Discrete Type Definition: Given a random experiment with an outcome space S, a function X that assigns to each element s in S one and only one real number x X(s) is called a random

More information

Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).

Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ). Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The

More information

Change of Continuous Random Variable

Change of Continuous Random Variable Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

More information

Monte Carlo Method: Probability

Monte Carlo Method: Probability John (ARC/ICAM) Virginia Tech... Math/CS 4414: The Monte Carlo Method: PROBABILITY http://people.sc.fsu.edu/ jburkardt/presentations/ monte carlo probability.pdf... ARC: Advanced Research Computing ICAM:

More information

Continuous Random Variables

Continuous Random Variables Probability 2 - Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)

More information

Notes on Distributions, Measures of Central Tendency, and Dispersion

Notes on Distributions, Measures of Central Tendency, and Dispersion Notes on Distributions, Measures of Central Tendency, and Dispersion Anthropological Sciences 192/292 Data Analysis in the Anthropological Sciences James Holland Jones & Ian G. Robertson February 1, 2006

More information

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

More information

Math 58. Rumbos Fall Solutions to Assignment #5

Math 58. Rumbos Fall Solutions to Assignment #5 Math 58. Rumbos Fall 2008 1 Solutions to Assignment #5 1. (Typographical Errors 1 ) Typographical and spelling errors can be either nonword errors or word errors. A nonword error is not a real word, as

More information

Empirical Rule Confidence Intervals Finding a good sample size. Outline. 1 Empirical Rule. 2 Confidence Intervals. 3 Finding a good sample size

Empirical Rule Confidence Intervals Finding a good sample size. Outline. 1 Empirical Rule. 2 Confidence Intervals. 3 Finding a good sample size Outline 1 Empirical Rule 2 Confidence Intervals 3 Finding a good sample size Outline 1 Empirical Rule 2 Confidence Intervals 3 Finding a good sample size -3-2 -1 0 1 2 3 Question How much of the probability

More information

2.8 Expected values and variance

2.8 Expected values and variance y 1 b a 0 y 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 a b (a) Probability density function x 0 a b (b) Cumulative distribution function x Figure 2.3: The probability density function and cumulative distribution function

More information

Applications of the Central Limit Theorem

Applications of the Central Limit Theorem Applications of the Central Limit Theorem October 23, 2008 Take home message. I expect you to know all the material in this note. We will get to the Maximum Liklihood Estimate material very soon! 1 Introduction

More information

Answers to some even exercises

Answers to some even exercises Answers to some even eercises Problem - P (X = ) = P (white ball chosen) = /8 and P (X = ) = P (red ball chosen) = 7/8 E(X) = (P (X = ) + P (X = ) = /8 + 7/8 = /8 = /9 E(X ) = ( ) (P (X = ) + P (X = )

More information

Introduction to Probability

Introduction to Probability Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

More information

University of California, Los Angeles Department of Statistics. Normal distribution

University of California, Los Angeles Department of Statistics. Normal distribution University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Normal distribution The normal distribution is the most important distribution. It describes

More information

Bivariate Distributions

Bivariate Distributions Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is

More information

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability

6-2 The Standard Normal Distribution. Uniform Distribution. Density Curve. Area and Probability. Using Area to Find Probability 6-2 The Standard Normal Distribution This section presents the standard normal distribution which has three properties: 1. Its graph is bell-shaped. 2. Its mean is equal to 0 (μ = 0). 3. Its standard deviation

More information

Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

More information

3. Continuous Random Variables

3. Continuous Random Variables 3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

More information

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8. Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

More information

TRANSFORMATIONS OF RANDOM VARIABLES

TRANSFORMATIONS OF RANDOM VARIABLES TRANSFORMATIONS OF RANDOM VARIABLES 1. INTRODUCTION 1.1. Definition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have

More information

Engineering 323 Beautiful Homework Set 7 1 of 7 Morelli Problem The cdf of checkout duration X as described in Exercise 1 is

Engineering 323 Beautiful Homework Set 7 1 of 7 Morelli Problem The cdf of checkout duration X as described in Exercise 1 is Engineering 33 Beautiful Homework Set 7 of 7 Morelli Problem.. The cdf of checkout duration X as described in Eercise is F( )

More information

Will Landau. Feb 26, 2013

Will Landau. Feb 26, 2013 ,, and,, and Iowa State University Feb 26, 213 Iowa State University Feb 26, 213 1 / 27 Outline,, and Iowa State University Feb 26, 213 2 / 27 of continuous distributions The p-quantile of a random variable,

More information

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1]. Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

More information

Review. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS

Review. Lecture 3: Probability Distributions. Poisson Distribution. May 8, 2012 GENOME 560, Spring Su In Lee, CSE & GS Lecture 3: Probability Distributions May 8, 202 GENOME 560, Spring 202 Su In Lee, CSE & GS suinlee@uw.edu Review Random variables Discrete: Probability mass function (pmf) Continuous: Probability density

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

This HW reviews the normal distribution, confidence intervals and the central limit theorem.

This HW reviews the normal distribution, confidence intervals and the central limit theorem. Homework 3 Solution This HW reviews the normal distribution, confidence intervals and the central limit theorem. (1) Suppose that X is a normally distributed random variable where X N(75, 3 2 ) (mean 75

More information

Lecture 10: Other Continuous Distributions and Probability Plots

Lecture 10: Other Continuous Distributions and Probability Plots Lecture 10: Other Continuous Distributions and Probability Plots Devore: Section 4.4-4.6 Page 1 Gamma Distribution Gamma function is a natural extension of the factorial For any α > 0, Γ(α) = 0 x α 1 e

More information

Introduction to Random Variables (RVs)

Introduction to Random Variables (RVs) ECE270: Handout 5 Introduction to Random Variables (RVs) Outline: 1. informal definition of a RV, 2. three types of a RV: a discrete RV, a continuous RV, and a mixed RV, 3. a general rule to find probability

More information

Regression Estimation - Least Squares and Maximum Likelihood. Dr. Frank Wood

Regression Estimation - Least Squares and Maximum Likelihood. Dr. Frank Wood Regression Estimation - Least Squares and Maximum Likelihood Dr. Frank Wood Least Squares Max(min)imization Function to minimize w.r.t. b 0, b 1 Q = n (Y i (b 0 + b 1 X i )) 2 i=1 Minimize this by maximizing

More information

PHP 2510 Central limit theorem, confidence intervals. PHP 2510 October 20,

PHP 2510 Central limit theorem, confidence intervals. PHP 2510 October 20, PHP 2510 Central limit theorem, confidence intervals PHP 2510 October 20, 2008 1 Distribution of the sample mean Case 1: Population distribution is normal For an individual in the population, X i N(µ,

More information

Probability and Statistics

Probability and Statistics CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

More information

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%).

1. Consider an untested batch of memory chips that have a known failure rate of 8% (yield = 92%). eview of Introduction to Probability and Statistics Chris Mack, http://www.lithoguru.com/scientist/statistics/review.html omework #2 Solutions 1. Consider an untested batch of memory chips that have a

More information

Conditional expectation

Conditional expectation A Conditional expectation A.1 Review of conditional densities, expectations We start with the continuous case. This is sections 6.6 and 6.8 in the book. Let X, Y be continuous random variables. We defined

More information

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ

More information

Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

More information

Joint Continuous Distributions

Joint Continuous Distributions Joint Continuous Distributions Statistics 11 Summer 6 Copright c 6 b Mark E. Irwin Joint Continuous Distributions Not surprisingl we can look at the joint distribution of or more continuous RVs. For eample,

More information

WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

More information

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

More information

Stat 704 Data Analysis I Probability Review

Stat 704 Data Analysis I Probability Review 1 / 30 Stat 704 Data Analysis I Probability Review Timothy Hanson Department of Statistics, University of South Carolina Course information 2 / 30 Logistics: Tuesday/Thursday 11:40am to 12:55pm in LeConte

More information

4: Probability. What is probability? Random variables (RVs)

4: Probability. What is probability? Random variables (RVs) 4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

More information

MAS108 Probability I

MAS108 Probability I 1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

More information

An Introduction to First-Order Logic

An Introduction to First-Order Logic Outline An Introduction to K. 1 1 Lane Department of Computer Science and Electrical Engineering West Virginia University Syntax, Semantics and Outline Outline 1 The Syntax of 2 of Expressions Model Different

More information

Joint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014

Joint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014 Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample

More information

Histograms and density curves

Histograms and density curves Histograms and density curves What s in our toolkit so far? Plot the data: histogram (or stemplot) Look for the overall pattern and identify deviations and outliers Numerical summary to briefly describe

More information

Normal approximation to the Binomial

Normal approximation to the Binomial Chapter 5 Normal approximation to the Binomial 5.1 History In 1733, Abraham de Moivre presented an approximation to the Binomial distribution. He later (de Moivre, 1756, page 242 appended the derivation

More information

Chapter 2. The Normal Distribution

Chapter 2. The Normal Distribution Chapter 2 The Normal Distribution Lesson 2-1 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve

More information

Probabilities and Random Variables

Probabilities and Random Variables Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

More information

Absolute Maxima and Minima

Absolute Maxima and Minima Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all

More information

Continuous Distributions, Mainly the Normal Distribution

Continuous Distributions, Mainly the Normal Distribution Continuous Distributions, Mainly the Normal Distribution 1 Continuous Random Variables STA 281 Fall 2011 Discrete distributions place probability on specific numbers. A Bin(n,p) distribution, for example,

More information

The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University

The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University The Normal Distribution Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Continuous Random

More information

THE ENTROPY OF THE NORMAL DISTRIBUTION

THE ENTROPY OF THE NORMAL DISTRIBUTION CHAPTER 8 THE ENTROPY OF THE NORMAL DISTRIBUTION INTRODUCTION The normal distribution or Gaussian distribution or Gaussian probability density function is defined by N(x; µ, σ) = /σ (πσ ) / e (x µ). (8.)

More information

Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

Random Variable: A function that assigns numerical values to all the outcomes in the sample space. STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

More information

3.4 The Normal Distribution

3.4 The Normal Distribution 3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

ECE302 Spring 2006 HW5 Solutions February 21, 2006 1

ECE302 Spring 2006 HW5 Solutions February 21, 2006 1 ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 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

More information

8.2 Confidence Intervals for One Population Mean When σ is Known

8.2 Confidence Intervals for One Population Mean When σ is Known 8.2 Confidence Intervals for One Population Mean When σ is Known Tom Lewis Fall Term 2009 8.2 Confidence Intervals for One Population Mean When σ isfall Known Term 2009 1 / 6 Outline 1 An example 2 Finding

More information

We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students: MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

More information

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

More information

Each exam covers lectures from since the previous exam and up to the exam date.

Each exam covers lectures from since the previous exam and up to the exam date. Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,

More information

3. Renewal Theory. Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times.

3. Renewal Theory. Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times. 3. Renewal Theory Definition 3 of the Poisson process can be generalized: Let X 1, X 2,..., iidf(x) be non-negative interarrival times. Set S n = n i=1 X i and N(t) = max {n : S n t}. Then {N(t)} is a

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.2 Homework Answers 5.29 An automatic grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125

More information

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite COURSE OUTLINE Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4 Hours: Lecture/Lab/Other 4 Lecture Co- or Pre-requisite MAT151 or MAT149 with a minimum

More information

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical distributions are Theoretical Distributions 1. Binomial distribution 2. Poisson distribution Discrete distribution

More information

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions) Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course

More information

Math 576: Quantitative Risk Management

Math 576: Quantitative Risk Management Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 4 Haijun Li Math 576: Quantitative Risk Management Week 4 1 / 22 Outline 1 Basics

More information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014 STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

More information