Moments of Random Variables

Slides:

Advertisements

Similar presentations

Special random variables Chapter 5 Some discrete or continuous probability distributions.

Advertisements

Some additional Topics. Distributions of functions of Random Variables Gamma distribution,  2 distribution, Exponential distribution.

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law.

Chapter 3 Some Special Distributions Math 6203 Fall 2009 Instructor: Ayona Chatterjee.

Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events.

Use of moment generating functions. Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function.

Review of Basic Probability and Statistics

Probability Densities

1 Engineering Computation Part 6. 2 Probability density function.

A random variable that has the following pmf is said to be a binomial random variable with parameters n, p The Binomial random variable.

2. Random variables  Introduction  Distribution of a random variable  Distribution function properties  Discrete random variables  Point mass  Discrete.

The moment generating function of random variable X is given by Moment generating function.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 4 Continuous Random Variables and Probability Distributions.

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

Jointly distributed Random variables

5-1 Two Discrete Random Variables Example Two Discrete Random Variables Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.

4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,

Distribution Function properties. Density Function – We define the derivative of the distribution function F X (x) as the probability density function.

Moment Generating Functions 1/33. Contents Review of Continuous Distribution Functions 2/33.

Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.

4.2 Variances of random variables. A useful further characteristic to consider is the degree of dispersion in the distribution, i.e. the spread of the.

Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution.

Moment Generating Functions

Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.

Use of moment generating functions 1.Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …).

Continuous Distributions The Uniform distribution from a to b.

Probability & Statistics I IE 254 Summer 1999 Chapter 4  Continuous Random Variables  What is the difference between a discrete & a continuous R.V.?

4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,

Expectation for multivariate distributions. Definition Let X 1, X 2, …, X n denote n jointly distributed random variable with joint density function f(x.

Stats Probability Theory Summary. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes.

Engineering Statistics - IE 261

Expectation. Let X denote a discrete random variable with probability function p(x) (probability density function f(x) if X is continuous) then the expected.

INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2011 Pearson Education, Inc. Chapter 16 Continuous Random.

IE 300, Fall 2012 Richard Sowers IESE. 8/30/2012 Goals: Rules of Probability Counting Equally likely Some examples.

Chapter 5a:Functions of Random Variables Yang Zhenlin.

Brief Review Probability and Statistics. Probability distributions Continuous distributions.

Jointly distributed Random variables Multivariate distributions.

Continuous Distributions

Basics of Multivariate Probability

4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,

Statistics Lecture 19.

ASV Chapters 1 - Sample Spaces and Probabilities

The Exponential and Gamma Distributions

Functions and Transformations of Random Variables

Chapter 4 Continuous Random Variables and Probability Distributions

Expectations of Random Variables, Functions of Random Variables

Chapter 4. Inference about Process Quality

Keller: Stats for Mgmt & Econ, 7th Ed

Appendix A: Probability Theory

PRODUCT MOMENTS OF BIVARIATE RANDOM VARIABLES

The distribution function F(x)

The Bernoulli distribution

Some Rules for Expectation

Moment Generating Functions

Suppose you roll two dice, and let X be sum of the dice. Then X is

Mean & Variance of a Distribution

Probability Review for Financial Engineers

Example Suppose X ~ Uniform(2, 4). Let . Find .

Continuous Probability Distributions

Lecture 5 b Faten alamri.

Random WALK, BROWNIAN MOTION and SDEs

Chapter 6 Some Continuous Probability Distributions.

Functions of Random variables

3. Random Variables Let (, F, P) be a probability model for an experiment, and X a function that maps every to a unique point.

6.3 Sampling Distributions

Lecture 14: Multivariate Distributions

Chapter 3 : Random Variables

Chapter 8 Estimation.

Continuous Distributions

Presentation transcript:

Moments of Random Variables The moment generating function

Examples The Binomial distribution (parameters p, n) The Poisson distribution (parameter l)

The Exponential distribution (parameter l) The Standard Normal distribution (m = 0, s = 1)

The Gamma distribution (parameters a, l) The Chi-square distribution (degrees of freedom n) (a = n/2, l = 1/2)

Properties of Moment Generating Functions mX(0) = 1

The log of Moment Generating Functions Let lX (t) = ln mX(t) = the log of the moment generating function

Thus lX (t) = ln mX(t) is very useful for calculating the mean and variance of a random variable

Examples The Binomial distribution (parameters p, n)

The Poisson distribution (parameter l)

The Exponential distribution (parameter l)

The Standard Normal distribution (m = 0, s = 1)

The Gamma distribution (parameters a, l)

The Chi-square distribution (degrees of freedom n)

Jointly distributed Random variables Multivariate distributions

Discrete Random Variables

The joint probability function; p(x,y) = P[X = x, Y = y]

Marginal distributions Conditional distributions

Continuous Random Variables

Definition: Two random variable are said to have joint probability density function f(x,y) if

Marginal distributions Conditional distributions

Independence

Definition: Two random variables X and Y are defined to be independent if if X and Y are discrete if X and Y are continuous Thus in the case of independence marginal distributions ≡ conditional distributions

The Multiplicative Rule for densities if X and Y are discrete if X and Y are continuous

Proof: This follows from the definition for conditional densities: Hence and The same is true for continuous random variables.

Example: Suppose that a rectangle is constructed by first choosing its length, X and then choosing its width Y. Its length X is selected form an exponential distribution with mean m = 1/l = 5. Once the length has been chosen its width, Y, is selected from a uniform distribution form 0 to half its length. Find the probability that its area A = XY is less than 4.

Solution:

xy = 4 y = x/2

This part can be evaluated This part may require Numerical evaluation