Discrete Random Variables: Expectation, Mean and Variance

Slides:



Advertisements
Similar presentations
Presentation on Probability Distribution * Binomial * Chi-square
Advertisements

ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 6 – More Discrete Random Variables Farinaz Koushanfar ECE Dept., Rice University Sept 10,
2 Discrete Random Variable. Independence (module 1) A and B are independent iff the knowledge that B has occurred does not change the probability that.
Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Follows Jim Pitman’s book: Probability Section.
Review of Basic Probability and Statistics
CSE 3504: Probabilistic Analysis of Computer Systems Topics covered: Moments and transforms of special distributions (Sec ,4.5.3,4.5.4,4.5.5,4.5.6)
Probability Distributions
1 Review of Probability Theory [Source: Stanford University]
P robability Important Random Variable Independent random variable Mean and variance 郭俊利 2009/03/23.
Probability and Statistics Review
Engineering Probability and Statistics - SE-205 -Chap 3 By S. O. Duffuaa.
A random variable that has the following pmf is said to be a binomial random variable with parameters n, p The Binomial random variable.
Probability Mass Function Expectation 郭俊利 2009/03/16
Continuous Random Variables and Probability Distributions
CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Special discrete distributions Sec
Week 51 Theorem For g: R  R If X is a discrete random variable then If X is a continuous random variable Proof: We proof it for the discrete case. Let.
1 Probability distribution Dr. Deshi Ye College of Computer Science, Zhejiang University
Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.
CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics, 2007 Instructor Longin Jan Latecki Chapter 7: Expectation and variance.
Winter 2006EE384x1 Review of Probability Theory Review Session 1 EE384X.
Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.
Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
Chapter 5: The Binomial Probability Distribution and Related Topics Section 1: Introduction to Random Variables and Probability Distributions.
One Random Variable Random Process.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Mistah Flynn.
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.
IE 300, Fall 2012 Richard Sowers IESE. 8/30/2012 Goals: Rules of Probability Counting Equally likely Some examples.
Expected values of discrete Random Variables. The function that maps S into S X in R and which is denoted by X(.) is called a random variable. The name.
Continuous Random Variables and Probability Distributions
Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Follows Jim Pitman’s book: Probability Section.
Chapter 6 Large Random Samples Weiqi Luo ( 骆伟祺 ) School of Data & Computer Science Sun Yat-Sen University :
Random Variables By: 1.
Discrete Random Variable Random Process. The Notion of A Random Variable We expect some measurement or numerical attribute of the outcome of a random.
Week 61 Poisson Processes Model for times of occurrences (“arrivals”) of rare phenomena where λ – average number of arrivals per time period. X – number.
Chapter Five The Binomial Probability Distribution and Related Topics
Expectations of Random Variables, Functions of Random Variables
Probability for Machine Learning
Probability Review for Financial Engineers
4A: Probability Concepts and Binomial Probability Distributions
Distributions and expected value
Sets and Probabilistic Models
Conditional Probability, Total Probability Theorem and Bayes’ Rule
Berlin Chen Department of Computer Science & Information Engineering
Further Topics on Random Variables: 1
Further Topics on Random Variables: Derived Distributions
Discrete Random Variables: Basics
Berlin Chen Department of Computer Science & Information Engineering
Further Topics on Random Variables: Covariance and Correlation
Sets and Probabilistic Models
Independence and Counting
Conditional Probability, Total Probability Theorem and Bayes’ Rule
Discrete Random Variables: Joint PMFs, Conditioning and Independence
Discrete Random Variables: Basics
Berlin Chen Department of Computer Science & Information Engineering
Further Topics on Random Variables: Derived Distributions
Experiments, Outcomes, Events and Random Variables: A Revisit
Conditional Probability, Total Probability Theorem and Bayes’ Rule
Discrete Random Variables: Expectation, Mean and Variance
Discrete Random Variables: Expectation, Mean and Variance
Sets and Probabilistic Models
Independence and Counting
Berlin Chen Department of Computer Science & Information Engineering
Independence and Counting
Berlin Chen Department of Computer Science & Information Engineering
Further Topics on Random Variables: Covariance and Correlation
Further Topics on Random Variables: Derived Distributions
Discrete Random Variables: Basics
Sets and Probabilistic Models
Conditional Probability, Total Probability Theorem and Bayes’ Rule
Continuous Random Variables: Basics
Presentation transcript:

Discrete Random Variables: Expectation, Mean and Variance Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference: - D. P. Bertsekas, J. N. Tsitsiklis, Introduction to Probability , Sections 2.3-2.4

Motivation (1/2) An Illustrative Example: Suppose that you spin the wheel times, and that is the number of times that the outcome (money) is (there are distinct outcomes, ) What is the amount of money that you “expect” to get “per spin”? The total amount received is The amount received per spin is

Motivation (2/2) If the number of spins is very large, and if we are willing to interpret probabilities as relative frequencies, it is reasonable to anticipate that comes up a fraction of times that is roughly equal to Therefore, the amount received per spin can be also represented as

Expectation The expected value (also called the expectation or the mean) of a random variable , with PMF , is defined by Can be interpreted as the center of gravity of the PMF (Or a weighted average, in proportion to probabilities, of the possible values of ) The expectation is well-defined if That is, converges to a finite value The mean us the point at which the sum of the torques (力矩) from the weights to its left is equal to the sum of the torques from the weights to its right.

Moments The n-th moment of a random variable is the expected value of a random variable (or the random variable , ) The 1st moment of a random variable is just its mean (or expectation) is termed as X raised to the power of n (or the nth power), or the nth power of X.

Expectations for Functions of Random Variables Let be a random variable with PMF , and let be a function of . Then, the expected value of the random variable is given by To verify the above rule Let , and therefore ?

Variance The variance of a random variable is the expected value of a random variable The variance is always nonnegative (why?) The variance provides a measure of dispersion of around its mean The standard derivation is another measure of dispersion, which is defined as (a square root of variance) Easier to interpret, because it has the same units as

An Example Example 2.3: For the random variable with PMF Discrete Uniform Random Variable

Properties of Mean and Variance (1/2) Let be a random variable and let where and are given scalars Then, If is a linear function of , then a linear function of How to verify it ?

Properties of Mean and Variance (2/2) 1

Variance in Terms of Moments Expression We can also express variance of a random variable as

An Example Example 2.4: Average Speech Versus Average Time. If the weather is good (with probability 0.6), Alice walks the 2 miles to class at a speed of V=5 miles per hour, and otherwise rides her motorcycle at a speech of V=30 miles per hour. What is the expected time E[T] to get to the class ?

Mean and Variance of the Bernoulli Example 2.5. Consider the experiment of tossing a biased coin, which comes up a head with probability and a tail with probability , and the Bernoulli random variable with PMF

Mean and Variance of the Discrete Uniform Consider a discrete uniform random variable with a nonzero PMF in the range [a, b]

Mean and Variance of the Poisson Consider a Poisson random variable with a PMF 1 1

Mean and Variance of the Binomial Consider a binomial random variable with a PMF 1 1

Mean and Variance of the Geometric Consider a geometric random variable with a PMF

An Example Example 2.3: The Quiz Problem. Consider a game where a person is given two questions and must decide which question to answer first Question 1 will be answered correctly with probability 0.8, and the person will then receive as prize $100 While question 2 will be answered correctly with probability 0.5, and the person will then receive as prize $200 If the first question attempted is answered incorrectly, the quiz terminates Which question should be answered first to maximize the expected value of the total prize money received?

Recitation SECTION 2.4 Expectation, Mean, Variance Problems 18, 19, 21, 24