Random Variate Generation R.B. Lenin (rblenin@daiict.ac.in) Autumn - - PowerPoint PPT Presentation

Random Variate Generation

R.B. Lenin (rblenin@daiict.ac.in) Autumn 2007

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 1 / 46

Outline

1

Random Variates Introduction General methods

2

Inverse method Inverse method – continuous case Uniform random variate Exponential random variate Weibull random variate Triangle random variate Inverse method – discrete case Geometric random variate

3

Convolution Method Binomial random variate Erlang random variate Poisson random variate

4

Composition Method Hyperexponential random variate

5

Acceptance-Rejection Method Gamma random variate Normal random variate Log-normal random variate

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 2 / 46

Random Variates Introduction

Random variates

The outcome of simulation of a random variable according to a given distribution law is called a random variate.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 3 / 46

Random Variates Introduction

Random variates

The outcome of simulation of a random variable according to a given distribution law is called a random variate. The procedure of simulating a random variable is called random variate generation.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 3 / 46

Random Variates Introduction

Random variates

The outcome of simulation of a random variable according to a given distribution law is called a random variate. The procedure of simulating a random variable is called random variate generation.

Random variate generation refers to the generation of variates whose probability distribution is different from that of the uniform on the interval [0, 1].

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 3 / 46

Random Variates General methods

General methods

Inverse transform method

Continuous case Discrete case

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 4 / 46

Random Variates General methods

General methods

Inverse transform method

Continuous case Discrete case

Convolution method

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 4 / 46

Random Variates General methods

General methods

Inverse transform method

Continuous case Discrete case

Convolution method Composition method

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 4 / 46

Random Variates General methods

General methods

Inverse transform method

Continuous case Discrete case

Convolution method Composition method Acceptance-rejection method

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 4 / 46

Random Variates General methods

General methods

Inverse transform method

Continuous case Discrete case

Convolution method Composition method Acceptance-rejection method Polar coordinate method

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 4 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case

Let U be a uniform random variable in [0, 1].

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 5 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case

Let U be a uniform random variable in [0, 1]. For any continuous distribution function F, the random variable X defined by X = F −1(U) has distribution F.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 5 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case

Let U be a uniform random variable in [0, 1]. For any continuous distribution function F, the random variable X defined by X = F −1(U) has distribution F. F −1(u) is defined to be that value of x such that F(x) = u.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 5 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Proof: Let FX denote the distribution function of X. We have to show that FX = F. FX(x) = Pr{X ≤ x} = Pr{F −1(U) ≤ x} = Pr{F(F −1(U) ≤ F(x)} = Pr{U ≤ F(x)} = F(x)

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 6 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Proof: Let FX denote the distribution function of X. We have to show that FX = F. FX(x) = Pr{X ≤ x} = Pr{F −1(U) ≤ x} = Pr{F(F −1(U) ≤ F(x)} = Pr{U ≤ F(x)} = F(x) A sample value of x of X is simulated as follows:

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 6 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Proof: Let FX denote the distribution function of X. We have to show that FX = F. FX(x) = Pr{X ≤ x} = Pr{F −1(U) ≤ x} = Pr{F(F −1(U) ≤ F(x)} = Pr{U ≤ F(x)} = F(x) A sample value of x of X is simulated as follows:

Generate a uniform random number r in [0, 1].

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 6 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Proof: Let FX denote the distribution function of X. We have to show that FX = F. FX(x) = Pr{X ≤ x} = Pr{F −1(U) ≤ x} = Pr{F(F −1(U) ≤ F(x)} = Pr{U ≤ F(x)} = F(x) A sample value of x of X is simulated as follows:

Generate a uniform random number r in [0, 1]. Set x = F −1(r)

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 6 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

The steps involved in using the inverse transform method in practice are

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 7 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

The steps involved in using the inverse transform method in practice are

1

Given:- The CDF FX(x) or the PDF fX(x):

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 7 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

The steps involved in using the inverse transform method in practice are

1

Given:- The CDF FX(x) or the PDF fX(x):

If fX is given, then first integrate it to get FX.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 7 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

The steps involved in using the inverse transform method in practice are

1

Given:- The CDF FX(x) or the PDF fX(x):

If fX is given, then first integrate it to get FX.

2

Generate a uniform random number r in [0, 1].

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 7 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

The steps involved in using the inverse transform method in practice are

1

Given:- The CDF FX(x) or the PDF fX(x):

If fX is given, then first integrate it to get FX.

2

Generate a uniform random number r in [0, 1].

3

Set FX(x) = r and solve for x.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 7 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Figure: Inverse method – continuous case

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 8 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Example Let X be a random variable having distribution function FX(x) = xn, 0 < x < 1.

R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 9 / 46

Inverse method Inverse method – continuous case

Inverse method – continuous case · · ·

Example Let X be a random variable having distribution function FX(x) = xn, 0 < x < 1. Let r be a uniform random number from [0, 1]. Then FX(x) = r ⇒ xn = r ⇒ x = r

1 n . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 9 / 46