Understanding Moments, Covariance, and Correlation in Continuous Probability Distributions, Slides of Statistical Physics

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Chapter 2

Topic: continuous pdf

Simulations in

Statistical Physics

Moments & Variance

The Central Moments

The Central moments of x are defined as

The second central moment has a particular meaning:

This is also called variance of x.

n i

n

p x x

g x x x



2 2

2 2

2 2 ( ) ( )





x x

p x x

x x p x x

i

i i

i

i i

i

2 2

var{ x }  x  x 

The standard deviation of x is    var{ x }

Continuous Random Variables

Consider the scattering of a photon by an atom. The angle at which the

photon is scattered has values that are continuous between 0 and 180o.

Given that x is a real continuous random variable that varies from

minus infinity to plus infinity. Then cumulative distribution is defined as

dx

dF x

f x

• F(x) = 0 for all x< 0. no sampling is done,

pdf = 0.

• F(x) = x, for all 0 < x < 1

pdf = 1 in this interval.

• F(x) = 0 , for all x >1,

pdf = 0.

F(x) = P{ a random selection of X gives a value less than }

The probability density function (pdf) may be defined as

0.0 0.5 1.0 1.5 2.

0.

0.

0.

0.

0.

1.

1.

1.

III

F (x) II

x

% How to generate uniform dis.

% The domain is generated

x = -1:0.1:11;

% now the pdf for x values

pdf = unifpdf(x, 0, 10);

cdf = unifcdf(x, 0, 10);

subplot(1,2,1),plot(x,pdf)

title('pdf')

xlabel('X'), ylabel('f(x)')

axis([-1 11 0 0.2])

axis square

subplot(1,2,2),plot(x,cdf)

title('cdf')

xlabel('X'), ylabel('f(x)')

axis([-1 11 0 1.1])

axis square

MATLAB Program to Generate Uniform pdf & cdf

Examples of Continuous Probability Distributions

UNIFORM DISTRIBUTION:

x a

x x a

F x x

 

  

 

1 ,

, 0

( ) 0 , 0

Average:  x  a / 2

Brief Calculations:

x a

a x a

F x x

Variance: var{ } / 12

2 x  a

4 12

( )

var{ }

2 2

0

2

2 2

a a x f x dx

x x x

a   

   



0.0 0.5 1.0 1.5 2.

0.

0.

0.

0.

0.

1.

1.

1.

III

F (x) II

x

0.0 0.5 1.0 1.5 2. 0.

0.

0.

0.

0.

1.

F'(x)

x

0 1 2 3 4 5

0.

0.

0.

0.

0.

1.0 F(x)

x

0 1 2 3 4 5

1.0 F'(x)

x

Exponential Probability Distribution Function

The average value:

The variance of x is

0 , 0

( ) 1 exp( ), 0

 

   

x

F x  x x

0 , 0

( ) exp( ), 0

 

   

x

F x   x x

 exp(  ) 1 /

( )

  

 





 







x x dx

x x f x dx

2

2 2

1 /

var{ } ( ) ( )

 

 

  

  









x x f x dx x f x dx

Cauchy or Lorentz Probability Distribution Function

The average value: improper integral diverges

The variance of x is infinite since the integral diverges no matter how it is

evaluated.



a

x F x

1 tan









x x f x dx

 









2 2 var{ x } x f ( x ) dx x f ( x ) dx

2 2

a x

a F x 