Consider a direct sampling simulation, the outcome is usually a sequence of values

which are different realizations of a random variable ξ , of distribution π(ξ). We assume that these values are independent and identically distributed (i.i.d.).

Using these N realisations, the best estimation of the mean

of the distribution π(ξ) of ξ is

Can we determine the statistical error associated with this estimation?

The mean XN being the average of N i.i.d. random variables, the central limit theorem should apply and the result writes:

Test this prediction with the following example: Sample N random numbers

taken from a uniform distribution on [0,1] and give an estimation of the average with its associated error.

Quality Control

We consider the direct sampling of the distribution

How can you sample this distribution?

Compute the average of ξ analytically.

Estimate the error using the previous method. Compare your results for different values of α. (First take α=3 and then try α=1.2).

To understand the difference between the different values of α let us introduce the "Quality control" method, which consists in plotting the running average defined as

import pylab,math, numpy,random
Ntime =10000random.seed(2)
alpha =3
xmeanexact = alpha / (alpha - 1.)
xicontrol =[]
xtot =0.
for itime inrange(Ntime):
xi ='think a little bit'
xtot += xi
xicontrol.append( xtot / float(itime + 1))
pylab.title('Quality Control, alpha = 3')
pylab.xlabel('time')
pylab.ylabel('running average of xi')time=[t for t inrange(Ntime)]
meanexact =[ xmeanexact for t inrange(Ntime)]
pylab.plot(xicontrol,'r-')
pylab.plot(time, meanexact,'k-')
pylab.axis([0, Ntime, xmeanexact * 0.9, xmeanexact * 1.1])
pylab.show()

Run this program for Ntime=10000, 50000, 250000 and compare your results for different values of α.

Compute analytically the variance of ξ as a function of α. Comment.

Stable distribution

Take Ntime=10,100,1000... Study the distribution of the variable

as N becomes large. Show that after a proper rescaling of this variable, the distribution converge (for large N) to a unique curve. What is this limit distribution?

import pylab,math, numpy,random
Nsample =20000
Ntime =100random.seed(2)
alpha =3
xmean = alpha/(alpha-1.)
Nfactor =float(Ntime)**0.5#(1. - 1./alpha)
xsum =[0. for n inrange(Nsample)]for isample inrange(Nsample):
xi =[0. for n inrange(Ntime)]for itime inrange(Ntime):
xi[itime]=random.random()**(-1./alpha)
xsum[isample]=(sum(xi)/float(Ntime) - xmean) * Nfactor
pylab.hist(xsum, bins=30,range=(-10,10),normed=True)
pylab.title('Example 3, Rescaled Distributions, alpha = 3')
pylab.xlabel('distribution')
pylab.ylabel('x')
pylab.show()

## Table of Contents

Class Session 04: Errors and fluctuations## Errors and Central limit theorem

Consider a direct sampling simulation, the outcome is usually a sequence of values

which are different realizations of a random variable

ξ, of distributionπ(ξ). We assume that these values areindependent and identically distributed(i.i.d.).Using these

Nrealisations, the best estimation of the meanof the distribution

π(ξ) ofξisCan we determine the statistical error associated with this estimation?

The mean

XNbeing the average ofNi.i.d. random variables, the central limit theorem should apply and the result writes:Test this prediction with the following example: Sample N random numbers

taken from a uniform distribution on [0,1] and give an estimation of the average with its associated error.

## Quality Control

We consider the direct sampling of the distribution- How can you sample this distribution?
- Compute the average of
- Estimate the error using the previous method. Compare your results for different values of

To understand the difference between the different values ofξanalytically.α. (First takeα=3 and then tryα=1.2).αlet us introduce the "Quality control" method, which consists in plotting the running average defined asα.ξas a function ofα. Comment.## Stable distribution

Take Ntime=10,100,1000... Study the distribution of the variable

as

Nbecomes large. Show that after a proper rescaling of this variable, the distribution converge (for largeN) to a unique curve. What is this limit distribution?[Print this page]