Stochastic Simulation Generation of random variables Discrete - PowerPoint PPT Presentation

Stochastic Simulation Generation of random variables Discrete sample space Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby – Denmark Email: bfn@imm.dtu.dk

Plan W1.1-2 Plan W1.1-2 Random number generation Independent, uniformly distributed RN Distribution Independent Random variable Model Output DTU 02443 – lecture 3 2

Random variables Random variables Aim • The scope is the generation of independent random variables X 1 , X 2 , ... X n with a given distribution , F x ( x ) , (or probability density function [pdf]). • We assume we have access to a supply ( U i ) of random numbers, independent samples from the uniform distribution on ]0 , 1[ . • Our task is to transform U i into X i . DTU 02443 – lecture 3 3

Uniform distribution I Uniform distribution I Our norm distribution or building block, U (0 , 1) f ( x ) = 1 F ( x ) = x for 0 ≤ x ≤ 1 E ( X ) = 1 1 Var ( X ) = 2 12 1 0 1 DTU 02443 – lecture 3 4

Coin Coin or uniform distribution �� 1 �� 0 �� 1 �� 1/2 0 1 X = 0 , 1 1/2 0 P ( X = i ) = 1 0 1 2 1/2 � � U > 1 X := X = ⌊ (2 U ) ⌋ 2 0 0 1 DTU 02443 – lecture 3 5

Bernoulli trial Bernoulli trial Toss a coin with P ( X = 1) = p and P ( X = 0) = 1 − p . F ( x ) = P ( X ≤ x ) X = U > 1 − p PDF X = 0 0 ≤ U ≤ 1 − p p 1−p X = 1 1 − p < U ≤ 1 0 1 0 1 1 CDF �� 1−p 0 1 �� 0 1 0 1−p 0 1 DTU 02443 – lecture 3 6

A fair die A fair die or uniform distribution 1 �� 1 �� 2 3 �� 5 �� 6 4 �� 0 1/6 1 1/6 X = 1 , 2 , ... 6 0 1 2 3 4 5 6 P ( X = i ) = 1 / 6 1/6 X = ⌊ (6 U ) ⌋ + 1 0 1 2 3 4 5 6 Can be generalized 6 → k . DTU 02443 – lecture 3 7

Discrete distribution - direct (crude) method Discrete distribution - direct (crude) method Suppose X can take k distinct values x 1 < x 2 < ... x k with p i = P ( X = x i ) , i = 1 , 2 , ... , k Then X takes the value x i with probability p i if U falls in an interval with length p i . That is if i − 1 i � � p j < U ≤ p j j =1 j =1 or X = x i if F ( x i − 1 ) < U ≤ F ( x i ) DTU 02443 – lecture 3 8

Geometric distribution, NB (1 , p ) Geometric distribution, NB (1 , p ) The discrete time version of waiting time. Memory-less. P ( X = n ) = (1 − p ) n − 1 p n = 1 , 2 , ... f ( n ) = P ( X ≤ n ) = 1 − (1 − p ) n F ( n ) = 1 − (1 − p ) n − 1 < U ≤ 1 − (1 − p ) n X = n if F ( n − 1) < U ≤ F ( n ) n − 1 < log (1 − U ) log (1 − p ) ≤ n �� log ( U ) X = + 1 log (1 − p ) DTU 02443 – lecture 3 9

Discrete distribution II Discrete distribution II �� 1 2 3 4 5 6 7 �� 1 0 P1 P2 1. Generate U 2. Find the interval i which U belong to. P i − 1 < U ≤ P i 3. output x i • Linear search (E ( X ) ) • Rearrangement of intervals • Binary search • Indexed search DTU 02443 – lecture 3 10

Rejection Method Rejection Method Simple rejection More optimistic: acceptance method. Assume P ( X = i ) = p i for i = 1 , 2 , ... k . p(i) Let c ≥ p i (then p i /c ≤ 1 ). C �� 1. I = ⌊ ( k ∗ U 1 ) ⌋ + 1 �� 2. if U 2 ≤ p I /c output: I �� Else goto 1. �� 1 2 3 4 1 p i k c frequency for i : = p i p j � k 1 j =1 k c DTU 02443 – lecture 3 11

Alias method Alias method • A method for generating discrete random variates of general type • From discrete uniform to general discrete • Generate one random number • One comparison • Potentially one table look-up • Drawback: Complex set-up procedure 02443 02443 – lecture 3 12

Stochastic Simulation Generation of random variables Discrete - PowerPoint PPT Presentation

Stochastic Simulation Generation of random variables Discrete sample space Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: bfn@imm.dtu.dk Plan W1.1-2 Plan W1.1-2

Stochastic Simulation Independent, uniformly distributed RN Generation of random variables

Stochastic Simulation Generation of random variables Continuous sample space Bo Friis Nielsen

Stochastic Simulation Random number generation Bo Friis Nielsen Applied Mathematics and Computer

Stochastic Simulation Random number generation Bo Friis Nielsen Applied Mathematics and Computer

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Random-Variate Generation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

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Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson & Nicol Discrete-Event

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f A ca mm b The probability that defined is Isa as E Prew prey Prata a WE Naka two rolled

Preliminaries 32 / 384 Random variables Let V = { V 1 , . . . , V n } , n 1 , be a set of

1 7 8 Types of simulation models Dynamic simulation models Stochastic vs. deterministic.

Stochastic Simulation Generation of random variables Discrete - PowerPoint PPT Presentation

Stochastic Simulation Generation of random variables Discrete sample space Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: bfn@imm.dtu.dk Plan W1.1-2 Plan W1.1-2

Stochastic Simulation Independent, uniformly distributed RN Generation of random variables

Stochastic Simulation Generation of random variables Continuous sample space Bo Friis Nielsen

Stochastic Simulation Random number generation Bo Friis Nielsen Applied Mathematics and Computer

Stochastic Simulation Random number generation Bo Friis Nielsen Applied Mathematics and Computer

Simulation Monte Carlo Monte Carlo simulation Outcome of a single stochastic simulation run

Comparison of Systems CS/ECE 541 1 1. Stochastic Ordering Let X and Y be random variables. We

Discrete Random Variables October 7, 2010 Discrete Random Variables Random Variables In many

CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science

E [ X ] = X 1 ( 2 ) ? X 1 ( 2 ) = { HHT , HTH , THH } . All the outcomes a Pr

Stochastic geometry and random generation 1 Stochastic geometry and random generation

Stochastic Processes Will Perkins March 7, 2013 Stochastic Processes Q: What is a Stochastic

6. random variables T T T T H T H H Random VariablesIntro 2

E [ X ] = X 1 ( a ) := { | X ( ) = a } . a Pr [ X = a ] . 3. Important

Random-Variate Generation Banks, Carson, Nelson &amp; Nicol Discrete-Event System Simulation

Random-Variate Generation Banks, Carson, Nelson &amp; Nicol Discrete-Event System Simulation

Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson &amp; Nicol Discrete-Event

Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson &amp; Nicol Discrete-Event

19. Series Representation of Stochastic Processes t Given information about a stochastic

An Introduction to Stochastic Simulation Stephen Gilmore Laboratory for Foundations of Computer

Imprecise random variables, random sets, and Monte Carlo simulation Thomas Fetz, Michael

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f A ca mm b The probability that defined is Isa as E Prew prey Prata a WE Naka two rolled

Preliminaries 32 / 384 Random variables Let V = { V 1 , . . . , V n } , n 1 , be a set of

1 7 8 Types of simulation models Dynamic simulation models Stochastic vs. deterministic.

Random-Variate Generation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Random-Variate Generation Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson & Nicol Discrete-Event

Chapter 5 Statistical Models in Simulation Banks, Carson, Nelson & Nicol Discrete-Event