Part II: Discrete Random Variables

Slides:

Advertisements

Similar presentations

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.

Advertisements

Chapter 2 Multivariate Distributions Math 6203 Fall 2009 Instructor: Ayona Chatterjee.

Continuous Random Variable (1). Discrete Random Variables Probability Mass Function (PMF)

Review of Basic Probability and Statistics

1 Def: Let and be random variables of the discrete type with the joint p.m.f. on the space S. (1) is called the mean of (2) is called the variance of (3)

Test 2 Stock Option Pricing

Probability Theory Part 2: Random Variables. Random Variables  The Notion of a Random Variable The outcome is not always a number Assign a numerical.

Probability Distributions – Finite RV’s Random variables first introduced in Expected Value def. A finite random variable is a random variable that can.

Assignment 2 Chapter 2: Problems  Due: March 1, 2004 Exam 1 April 1, 2004 – 6:30-8:30 PM Exam 2 May 13, 2004 – 6:30-8:30 PM Makeup.

Samples vs. Distributions Distributions: Discrete Random Variable Distributions: Continuous Random Variable Another Situation: Sample of Data.

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 Distributions Random Variables: Finite and Continuous A review MAT174, Spring 2004.

Probability Distributions Random Variables: Finite and Continuous Distribution Functions Expected value April 3 – 10, 2003.

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

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.

Probability Distributions: Finite Random Variables.

LECTURE UNIT 4.3 Normal Random Variables and Normal Probability Distributions.

Pairs of Random Variables Random Process. Introduction  In this lecture you will study:  Joint pmf, cdf, and pdf  Joint moments  The degree of “correlation”

Biostatistics Lecture 7 4/7/2015. Chapter 7 Theoretical Probability Distributions.

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

Binomial Distributions Calculating the Probability of Success.

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}.

Chapter 5: Random Variables and Discrete Probability Distributions

Lecture 10: Continuous RV Probability Theory and Applications Fall 2005 September 29 Everything existing in the universe is the fruit of chance. Democritus.

1 Continuous Probability Distributions Continuous Random Variables & Probability Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering.

 Random variables can be classified as either discrete or continuous.  Example: ◦ Discrete: mostly counts ◦ Continuous: time, distance, etc.

Chapter 3 Discrete Random Variables and Probability Distributions  Random Variables.2 - Probability Distributions for Discrete Random Variables.3.

Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Chapter 3 Random Variables and Probability Distributions.

Chapter 3 Discrete Random Variables and Probability Distributions  Random Variables.2 - Probability Distributions for Discrete Random Variables.3.

Continuous Random Variable (1) Section Continuous Random Variable What is the probability that X is equal to x?

CSE 474 Simulation Modeling | MUSHFIQUR ROUF CSE474:

CHAPTER Discrete Models  G eneral distributions  C lassical: Binomial, Poisson, etc Continuous Models  G eneral distributions 

7.2 Means & Variances of Random Variables AP Statistics.

Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Chapter 3 Random Variables and Probability Distributions.

The Birthday Problem. The Problem In a group of 50 students, what is the probability that at least two students share the same birthday?

Part II: Discrete Random Variables 1.

Unit 4 Review. Starter Write the characteristics of the binomial setting. What is the difference between the binomial setting and the geometric setting?

Chapter 5 Joint Probability Distributions and Random Samples  Jointly Distributed Random Variables.2 - Expected Values, Covariance, and Correlation.3.

11.3 CONTINUOUS RANDOM VARIABLES. Objectives: (a) Understand probability density functions (b) Solve problems related to probability density function.

Random Variables By: 1.

Chapter 9: Joint distributions and independence CIS 3033.

Random Variable 2013.

Continuous Random Variable

Random Variables and Probability Distribution (2)

Cumulative distribution functions and expected values

ETM 607 – Spreadsheet Simulations

The distribution function F(x)

Chapter 4 Continuous Random Variables and Probability Distributions

Conditional Probability on a joint discrete distribution

AP Statistics: Chapter 7

Chapter 3 Discrete Random Variables and Probability Distributions

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

Probability Review for Financial Engineers

x1 p(x1) x2 p(x2) x3 p(x3) POPULATION x p(x) ⋮ Total 1 “Density”

ASV Chapters 1 - Sample Spaces and Probabilities

ASV Chapters 1 - Sample Spaces and Probabilities

... DISCRETE random variables X, Y Joint Probability Mass Function y1

Chapter 14 Monte Carlo Simulation

Cumulative Distribution Function

ASV Chapters 1 - Sample Spaces and Probabilities

ASV Chapters 1 - Sample Spaces and Probabilities

Chapter 3 : Random Variables

M248: Analyzing data Block A UNIT A3 Modeling Variation.

Chapter 2. Random Variables

The Geometric Distributions

Experiments, Outcomes, Events and Random Variables: A Revisit

Part II: Discrete Random Variables

1/2555 สมศักดิ์ ศิวดำรงพงศ์

Presentation transcript:

Part II: Discrete Random Variables http://neveryetmelted.com/categories/mathematics/

Chapter 6: Random Variables; Discrete Versus Continuous http://math.sfsu.edu/beck/quotes.html

Chapter 7: Probability Mass Functions and Cumulative Distribution Functions The 50-50-90 rule: Anytime You have a 50-50 chance of getting something right, there’s a 90% probability you’ll get it wrong. Andy Rooney http://brownsharpie.courtneygibbons.org/?p=161

Probability Mass Density Cumulative Density Function Def. 7.1: If X is a random variable, the probability that X is exactly equal to X is called the PMF pX(x) = P(X = x) Def. 7.2: If X is a random variable, the probability that X does not exceed x is called the CDF FX(x) = P(X ≤ x)

Histogram: interpretation of PMF Simulated 1000 times Theoretical Simulated 10,000 times

Example 7.4 (Fig. 7.3) mass CDF

More on CDFs CDFs should always be written as piece-wise functions like the following which is from Problem 2 on the handout 𝐹 𝑋 𝑥 = 0 𝑥≤0 0.369 0≤𝑥<1 0.584 0.816 1 1≤𝑥<2 2≤𝑥<3 03≤𝑥

More on CDFs (cont.) Because CDFs are not right continuous, P(X ≤ a) ≠ P(X < a) FX(a) = P(X ≤ a) FX(a-) = P(X < a) To calculate a probability P(a < X ≤ b) = FX(b) – FX(a) P(a ≤ X < b) = FX(b-) – FX(a-) a b a b

Calculation of Probabilities from CDFs Let X be a random variable. Then for all real numbers a,b where a < b P(a < X ≤ b) = FX(b) – FX(a) P(a ≤ X ≤ b) = FX(b) – FX(a-) P(a < X < b) = FX(b-) – FX(a) P(a ≤ X < b) = FX(b-) – FX(a-) P(X = a) = FX(a) – FX(a-)

Chapter 8: Jointly Distributed Random Variables; Independence and Conditioning The most misleading assumptions are the ones you don’t even know you’re making. Douglas Adams and Mark Carwardine http://math.stackexchange.com/questions/314072/joint-probability-mass-function X

Chapters 9: Expected Values of Discrete Random Variables http://www.cartoonstock.com/directory/a/average_family_gifts.asp X

Chapter 10: Expected Values of Sums of Random Variables http://faculty.wiu.edu/JR-Olsen/wiu/stu/m206/front.htm X

Chapter 11: Expected Values of Functions of Discrete Random Variables; Variance of Discrete Random Variables http://fightingdarwin.blogspot.com/2011_12_01_archive.html X