Computational Physics - Department of Physics

352 11 Outline of the Monte Carlo Strategy

The expectation value of a quantityf(x)is then given by for example

〈f〉=

∫b

a

f(x)p(x)dx.

We have already seen the use of the last equation when we applied the crude Monte Carlo
approach to the evaluation of an integral.
There are at least three PDFs which one may encounter. These are the

1. uniform distribution
   p(x) =

1

b−a

Θ(x−a)Θ(b−x),

yielding probabilities different from zero in the interval[a,b]. The mean value and the

variance for this distribution are discussed in section 11.3.

2. The exponential distribution
   p(x) =αexp(−αx),
   yielding probabilities different from zero in the interval[ 0 ,∞)and with mean value

μ=

∫∞

0

xp(x)dx=

∫∞

0

xαexp(−αx)dx=^1

α

and variance

σ^2 =

∫∞

0

x^2 p(x)dx−μ^2 =

1

α^2.

3. Finally, we have the so-called univariate normal distribution, or just the normal distribu-
   tion
   p(x) =

1

b

√

2 π

exp

(

−

(x−a)^2

2 b^2

)

with probabilities different from zero in the interval(−∞,∞). The integral

∫∞

−∞exp

(

−(x^2

)

dx

appears in many calculations, its value is

√

π, a result we will need when we compute the

mean value and the variance. The mean value is

μ=

∫∞

0

xp(x)dx=

1

b

√

2 π

∫∞

−∞

xexp

(

−

(x−a)^2

2 b^2

)

dx,

which becomes with a suitable change of variables

μ=

1

b

√

2 π

∫∞

−∞

b

√

2 (a+b

√

2 y)exp−y^2 dy=a.

Similarly, the variance becomes

σ^2 =^1

b

√

2 π

∫∞

−∞

(x−μ)^2 exp

(

−(x−a)

2

2 b^2

)

dx,

and inserting the mean value and performing a variable change we obtain

σ^2 =

1

b

√

2 π

∫∞

−∞

b

√

2 (b

√

2 y)^2 exp

(

−y^2

)

dy=

2 b^2

√

π

∫∞

−∞

y^2 exp

(

−y^2

)

dy,

and performing a final integration by parts we obtain the well-known resultσ^2 =b^2. It is

useful to introduce the standard normal distribution as well, defined byμ=a= 0 , viz. a

distribution centered around zero and with a varianceσ^2 = 1 , leading to

p(x) =

1

√

2 π

exp

(

−

x^2

2

)

. (11.7)