Distribution Gamma Function Stochastic Process Tutorial 4, STAT1301 Fall 2010, 12OCT2010, By Joseph Dong.

Slides:

Advertisements

Similar presentations

In-Class Exercise: The Poisson Distribution

Advertisements

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

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

Many useful applications, especially in queueing systems, inventory management, and reliability analysis. A connection between discrete time Markov chains.

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

Random Variable Tutorial 3 STAT1301 Fall OCT2010, By Joseph Dong.

Discrete Uniform Distribution

MATH 1107 Introduction to Statistics

Tutorial 10, STAT1301 Fall 2010, 30NOV2010, By Joseph Dong.

Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Follows Jim Pitman’s book: Probability Section.

MATH408: Probability & Statistics Summer 1999 WEEK 4 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering.

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Discrete random variables Probability mass function Distribution function (Secs )

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

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Special discrete distributions Sec

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Stochastic processes Bernoulli and Poisson processes (Sec. 6.1,6.3.,6.4)

Elementary Distribution Theory: Transforming a Bivariate Distribution Tutorial 9, STAT1301 Fall 2010, 23NOV2010, By Joseph Dong.

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

Tutorial 6, STAT1301 Fall 2010, 02NOV2010, By Joseph Dong RANDOM V ECTOR.

One Sample  M ean μ, Variance σ 2, Proportion π Two Samples  M eans, Variances, Proportions μ1 vs. μ2 σ12 vs. σ22 π1 vs. π Multiple.

Chapter 5 Some Discrete Probability Distributions.

Chapter 17 Probability Models math2200. I don’t care about my [free throw shooting] percentages. I keep telling everyone that I make them when they count.

Distributions Normal distribution Binomial distribution Poisson distribution Chi-square distribution Frequency distribution

1 Bernoulli trial and binomial distribution Bernoulli trialBinomial distribution x (# H) 01 P(x)P(x)P(x)P(x)(1 – p)p ?

Random Variables and Stochastic Processes –

Poisson Random Variable Provides model for data that represent the number of occurrences of a specified event in a given unit of time X represents the.

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

Discrete Probability Distributions. Random Variable Random variable is a variable whose value is subject to variations due to chance. A random variable.

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 5 Discrete Random Variables.

McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 5 Discrete Random Variables.

COMP 170 L2 L17: Random Variables and Expectation Page 1.

Probability Distributions u Discrete Probability Distribution –Discrete vs. continuous random variables »discrete - only a countable number of values »continuous.

Chapter Six Discrete Probability Distributions Section 6.3 The Poisson Probability Distribution.

Pemodelan Kualitas Proses Kode Matakuliah: I0092 – Statistik Pengendalian Kualitas Pertemuan : 2.

Math b (Discrete) Random Variables, Binomial Distribution.

Probability Refresher COMP5416 Advanced Network Technologies.

Statistics 3502/6304 Prof. Eric A. Suess Chapter 4.

Topic 3 - Discrete distributions Basics of discrete distributions - pages Mean and variance of a discrete distribution - pages ,

4.3 More Discrete Probability Distributions NOTES Coach Bridges.

Random Variables Example:

Chapter 4. Random Variables - 3

Chapter 3 Statistical Models or Quality Control Improvement.

Chapter 31Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012 John Wiley & Sons, Inc.

Probability Generating Functions Suppose a RV X can take on values in the set of non-negative integers: {0, 1, 2, …}. Definition: The probability generating.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Discrete Probability Distributions 6.

2.2 Discrete Random Variables 2.2 Discrete random variables Definition 2.2 –P27 Definition 2.3 –P27.

Discrete Probability Distributions Chapter 4. § 4.3 More Discrete Probability Distributions.

Probability and Statistics Dr. Saeid Moloudzadeh Poisson Random Variable 1 Contents Descriptive Statistics Axioms of Probability Combinatorial.

Created by Tom Wegleitner, Centreville, Virginia Section 4-5 The Poisson Distribution.

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

Review of Probability Theory

Statistical Quality Control, 7th Edition by Douglas C. Montgomery.

Math 4030 – 4a More Discrete Distributions

Probability Distributions: a review

Discrete Probability Distributions

STAT 312 Chapter 7 - Statistical Intervals Based on a Single Sample

1200 Hz = 1200/sec would mean in 5 minutes we

Multinomial Distribution

Probability & Statistics Probability Theory Mathematical Probability Models Event Relationships Distributions of Random Variables Continuous Random.

Discrete Probability Distribution

Some Discrete Probability Distributions

Useful Discrete Random Variable

CHAPTER 6 Statistical Inference & Hypothesis Testing

Chapter 4 Discrete Probability Distributions.

Chapter 3 : Random Variables

Bernoulli Trials Two Possible Outcomes Trials are independent.

Random Variables A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment. A random variable can be.

Each Distribution for Random Variables Has:

Statistical Models or Quality Control Improvement

Distribution Function of Random Variables

Presentation transcript:

Distribution Gamma Function Stochastic Process Tutorial 4, STAT1301 Fall 2010, 12OCT2010, By Joseph Dong

Reference 2

Recall: Distribution of a Random Variable 3

4

5

Handout Problems 6 & 7 Problem 6: ▫Gamma function and integration practice Problem 7: ▫important continuous distributions and their relationships 6 Technical

From Bernoulli Trials to Discrete Waiting Time (Handout Problems 1-4) 7

Poisson [ pwa’s ɔ ̃] Distribution Poisson Approximation to Binomial (PAB) ▫Handout Problem 5 The true utility of Poisson distribution—Poisson process: ▫Sort of the limiting case of Bernoulli trials (use PAB to facilitate thinking) ▫“continuous” Bernoulli trials 8

Sequence of Random Variables 9

Stochastic Process: Discrete-time and Continuous-time 10

Discrete-time processContinuous-time process 11

Bernoulli Trials (Bernoulli Process) 12

Poisson Process 13

Discrete Distribution Based On Bernoulli Trails 14

Continuous Distribution Based On Poisson Process 15

Examples of Poisson Process Radioactive disintegrations Flying-bomb hits on London Chromosome interchanges in cells Connection to wrong number Bacteria and blood counts Feller: An Introduction to Probability Theory and Its Applications (3e) Vol. 1. §VI.6. 16

Radioactive Disintegrations 17 Geiger Counter Geiger Rutherford Chadwick

Rutherford, Chadwick, and Ellis’ 1920 Experiment #intervals (recorded) N 2608 Intensity (7.5s)

Explanation There are 57 time intervals (7.5 sec each) recorded zero emission. There are 203 time intervals (7.5 sec each) recorded 1 emission. …… There are total 2608 time intervals (7.5 sec each) involved. On average, each interval recorded 3.87 emissions. Use 3.87 as the intensity of the Poisson process that models the counts of emissions on each of the 2608 intervals. 19

What’s the waiting time until recording 40 emissions? 20