Computational Physics - Department of Physics

354 11 Outline of the Monte Carlo Strategy

p(x) =

λx

x!

e−λ x= 0 , 1 ,...,;λ> 0.

In this case both the mean value and the variance are easier tocalculate,

μ=

∞

∑

x= 0

x

λx

x!

e−λ=λe−λ

∞

∑

x= 1

λx−^1

(x− 1 )!

=λ,

and the variance isσ^2 =λ. An example of applications of the Poisson distribution could be the
counting of the number ofα-particles emitted from a radioactive source in a given timeinter-
val. In the limit ofn→∞and for small probabilitiesy, the binomial distribution approaches
the Poisson distribution. Settingλ=ny, withythe probability for an event in the binomial
distribution we can show that

nlim→∞

(

n

x

)

yx( 1 −y)n−xe−λ=

∞

∑

x= 1

λx

x!e

−λ,

see for example Refs. [63,64] for a proof.

11.2.1Multivariable Expectation Values

An important quantity is the so called covariance, a variantof the variance. Consider the
set{Xi}ofnstochastic variables (not necessarily uncorrelated) withthe multivariate PDF
P(x 1 ,...,xn). Thecovarianceof two of the stochastic variables,XiandXj, is defined as follows

Cov(Xi,Xj)≡

〈

(xi−〈xi〉)(xj−〈xj〉)

〉

=

∫

···

∫

(xi−〈xi〉)(xj−〈xj〉)P(x 1 ,...,xn)dx 1 ...dxn (11.8)

with
〈xi〉=

∫

···

∫

xiP(x 1 ,...,xn)dx 1 ...dxn

If we consider the above covariance as a matrixCi j=Cov(Xi,Xj), then the diagonal elements
are just the familiar variances,Cii=Cov(Xi,Xi) =Var(Xi). It turns out that all the off-diagonal
elements are zero if the stochastic variables are uncorrelated. This is easy to show, keeping in
mind the linearity of the expectation value. Consider the stochastic variablesXiandXj, (i 6 =j)

Cov(Xi,Xj) =

〈

(xi−〈xi〉)(xj−〈xj〉)

〉

=〈xixj−xi〈xj〉−〈xi〉xj+〈xi〉〈xj〉〉

=〈xixj〉−〈xi〈xj〉〉−〈〈xi〉xj〉+〈〈xi〉〈xj〉〉

=〈xixj〉−〈xi〉〈xj〉−〈xi〉〈xj〉+〈xi〉〈xj〉

=〈xixj〉−〈xi〉〈xj〉

IfXiandXjare independent, we get〈xixj〉=〈xi〉〈xj〉, resulting inCov(Xi,Xj) = 0 (i 6 =j).
Also useful for us is the covariance of linear combinations of stochastic variables. Let{Xi}
and{Yi}be two sets of stochastic variables. Let also{ai}and{bi}be two sets of scalars.
Consider the linear combination

U=∑

i

aiXi V=∑

j

bjYj