Various Random Number Generators and the Applications

Slides:

Advertisements

Similar presentations

What is the standard deviation good for? Numerical data that come from an experiment carry inherently some “unit of measure”, for example How long a battery.

Advertisements

Kurtis Cahill James Badal.  Introduction  Model a Maze as a Markov Chain  Assumptions  First Approach and Example  Second Approach and Example 

Module C9 Simulation Concepts. NEED FOR SIMULATION Mathematical models we have studied thus far have “closed form” solutions –Obtained from formulas --

7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These.

Monte Carlo Methods in Partial Differential Equations.

Introduction to Monte-carlo Analysis for software development

Lesson Confidence Intervals about a Population Standard Deviation.

Sensor Positioning in Wireless Ad-hoc Sensor Networks Using Multidimensional Scaling Xiang Ji and Hongyuan Zha Dept. of Computer Science and Engineering,

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 1 Chapter 4 Numerical Methods for Describing Data.

1 Theoretical Physics Experimental Physics Equipment, Observation Gambling: Cards, Dice Fast PCs Random- number generators Monte- Carlo methods Experimental.

Key Concepts, continued To determine the probability of an outcome using continuous data, we use the proportion of the area under the normal curve associated.

7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These.

Monte Carlo Methods Versatile methods for analyzing the behavior of some activity, plan or process that involves uncertainty.

Hit-and-Miss (or Rejection) Monte Carlo Method:

TobiasEcon 472 Law of Large Numbers (LLN) and Central Limit Theorem (CLT)

July, 2000Guang Jin Statistics in Applied Science and Technology Chapter 7 - Sampling Distribution of Means.

Cognitive Radio for Dynamic Spectrum Allocation Systems Xiaohua (Edward) Li and Juite Hwu Department of Electrical and Computer Engineering State University.

Statistics What is statistics? Where are statistics used?

1 ENGINEERING MEASUREMENTS Prof. Emin Korkut. 2 Statistical Methods in Measurements.

EXCEL CHAPTER 6 ANALYZING DATA STATISTICALLY. Analyzing Data Statistically Data Characteristics Histograms Cumulative Distributions Classwork: 6.1, 6.6,

Sampling Distribution of the Sample Mean

Lesson 8: Basic Monte Carlo integration

Estimating the Value of a Parameter Using Confidence Intervals

Equations of Circles.

Circles in the Coordinate Plane

Circles Objectives: Write the Standard Form and General Form of the Equation of a Circle Find the Center and Radius of a Circle in Standard Form and General.

Cases and controls A case is an individual with a disease, whose location can be represented by a point on the map (red dot). In this table we examine.

Review Ways to “see” data Measures of central tendency

Optimization of Monte Carlo Integration

Joint Probability Distributions and Random Samples

Chapter 9 One and Two Sample Estimation

4. Numerical Integration

Equations of Circles.

7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These.

The Normal Distribution

Distributed Ray Tracing

3D Graphics Rendering PPT By Ricardo Veguilla.

Path Coupling And Approximate Counting

Random numbers What does it mean for a number to be random?

Equations of Circles.

Equations of Circles.

Department of Computer Science University of York

Volume 28, Issue 7, Pages e5 (April 2018)

CENTRAL LIMIT THEOREM specifies a theoretical distribution

Distributed Ray Tracing

Section 1.9 Distance and Midpoint Formulas; Circles

Lesson #9: Applications of a Normal Distribution

Lesson 1 CCSS Understand and apply theorems about circles.

Estimating the Value of a Parameter

Section 1.5 Circles Copyright © 2013 Pearson Education, Inc. All rights reserved.

Circles Objectives: Write the Standard Form and General Form of the Equation of a Circle Find the Center and Radius of a Circle in Standard Form and General.

Random WALK, BROWNIAN MOTION and SDEs

Continuous Random Variables

Nur Hasan Mahmud Shahen Lecturer of Mathematics. Department of Computer Science & Engineering (CSE). University- Institute of Science Trade & Technology,

Physics 2102 Lecture: 07 WED 28 JAN

Equations of Circles Part b.

Standard Equation of a Circle Definition of a Circle

Sampling: How to Select a Few to Represent the Many

Learning Target #21 Equations of Circles.

Introduction to Sampling Distributions

DESIGN OF EXPERIMENT (DOE)

Circles in the Coordinate Plane

Introduction to Probability: Solutions for Quizzes 4 and 5

Random numbers What does it mean for a number to be random?

Standard Equation of a Circle Definition of a Circle

Distributed Ray Tracing

Circles Objectives: Write the Standard Form and General Form of the Equation of a Circle Find the Center and Radius of a Circle in Standard Form and General.

Chapter Equations of Circles.

Find the Distribution Chapter 2.3.

Presentation transcript:

Various Random Number Generators and the Applications Jeff Atkinson Hirophysics.com

The advantages of using random numbers To make complicated calculation simple To estimate the result efficiently To simulate nature’s randomness Hirophysics.com

How to find  with random numbers Throw darts randomly to the target shown. Count how many darts placed within the circle. Calculate the percentage that gives the area of the circle. Divide it by r2 to obtain . r r Hirophysics.com

Case 1 The number of darts inside the shaded area = 6 The total number of darts = 8 The area of full circle = 46/8=3 If r=1, the above is also . Hirophysics.com

Case 2 The number of darts inside the shaded area = 12 The total number of darts = 17 The area of full circle = 412/17=2.824 If r=1, the above is also . Hirophysics.com

Case 3 The number of darts inside the shaded area = 47,000 The total number of darts = 60,000 The area of full circle = 447,000/60,000=3.1 3333 If r=1, the above is also . Hirophysics.com

Uniformly distributed random numbers Hirophysics.com

Box-Muller Distribution All data is centered around an average value and most data stays within a certain radius called the standard deviation (σ). To create a Box-Muller distribution, the following equations are applied to two uniformly distributed random numbers, U1 and U2: Hirophysics.com

Arcsine “Bucket” Distribution Most data comes together the upper and lower bounds. The numbers are from logistic map. Hirophysics.com

Computer simulation with 10,000 uniform random numbers The approximated  = 3.1356000000000002 Hirophysics.com

 with Box-Muller  with different standard deviations STDV ----  0.1 ---- 4.000 0.2 ---- 3.992 0.3 ---- 3.752 0.4 ---- 3.157 0.5 ---- 2.522 0.6 ---- 2.002 Hirophysics.com

Random Walk Mathematical determination of a path that is generated by a succession of random values. Simulation of Brownian motion and other natural random motions A formalism that are applied to other complex science and engineering Hirophysics.com

Random Walk with Uniform Random Number Eventual Average Distance = N Hirophysics.com

Random Walk with Box-Muller Eventual Average Distance = N Hirophysics.com

Random Walk with Arcsine Eventual Average Distance = 1.617N N --- Distance 500 --- 35.720 1500 --- 61.054 2500 --- 82.088 3500 --- 87.199 4500 --- 103.623 5500 --- 118.542 6500 --- 124.039 7500 --- 135.237 . 10,000 --- 162.129 Hirophysics.com

Conclusions For mathematical problems, such as , the data must be truly random and uniform. The Box-Muller distribution does not work to find  because it depends on the standard deviation. The random walk with arcsine distribution with logistic map [yn+1=4.0yn(1-yn)] gives the eventual average distance as 1.617N computationally. Hirophysics.com