Computational Physics - Department of Physics

11.1 Introduction 339

Eleven new attempts may results in a totally different sequence of numbers and so forth.
Repeating this exercise the next evening, will most likely never give you the same sequences.
Thus, we say that the outcome of this hobby of ours is truly random.
Random variables are hence characterized by a domain which contains all possible values
that the random value may take. This domain has a corresponding PDF.
To give you another example of possible random number spare time activities, consider
the radioactive decay of anα-particle from a certain nucleus. Assume that you have at your
disposal a Geiger-counter which registers every 10 ms whether anα-particle reaches the
counter or not. If we record a hit as 1 and no observation as zero, and repeat this experiment
for a long time, the outcome of the experiment is also truly random. We cannot form a specific
pattern from the above observations. The only possibility to say something about the outcome
is given by the PDF, which in this case is the well-known exponential function

λexp−(λx),

withλbeing proportional to the half-life of the given nucleus which decays.
If you wish to read more about the more formal aspects of MonteCarlo methods, see for
example Refs. [63–65].

11.1.1Definitions

Random numbers as we use them here are numerical approximations to the statistical con-
cept of stochastic variables, sometimes just called randomvariables. To understand the
behavior of pseudo random numbers we must first establish thetheoretical framework of
stochastic variables. Although this is typical textbook material, the nomenclature may differ
from one textbook to another depending on the level of difficulty of the book. We would there-
fore like to establish a nomenclature suitable for our purpose, one that we are going to use
consequently throughout this text.
A stochastic variable can be either continuous or discrete.In any case, we will denote
stochastic variables by capital lettersX,Y,...
There are two main concepts associated with a stochastic variable. Thedomainis the set
D={x}of all accessible values the variable can assume, so thatX∈D. An example of a
discrete domain is the set of six different numbers that we may get by throwing of a dice,
x∈{ 1 , 2 , 3 , 4 , 5 , 6 }.
Theprobability distribution function (PDF)is a functionp(x)on the domain which, in the
discrete case, gives us the probability or relative frequency with which these values ofX
occur
p(x) =Prob(X=x)

In the continuous case, the PDF does not directly depict the actual probability. Instead we
define the probability for the stochastic variable to assumeany value on an infinitesimal
interval aroundxto be p(x)dx. The continuous functionp(x)then gives us thedensityof
the probability rather than the probability itself. The probability for a stochastic variable to
assume any value on a non-infinitesimal interval[a,b]is then just the integral

Prob(a≤X≤b) =

∫b

a

p(x)dx

Qualitatively speaking, a stochastic variable representsthe values of numbers chosen as if
by chance from some specified PDF so that the selection of a large set of these numbers
reproduces this PDF.