Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.4 RANDOM VARIABLES AND DISTRIBUTIONS

Substituting this expression into (30.37) gives

W{ni}=N!

∏R

i=1

gi!

ni!(gi−ni)!

.

Such a system of particles has the names of no famous scientists attached to it, since it
appears that it never occurs in nature.

30.4 Random variables and distributions

Suppose an experiment has an outcome sample spaceS. A real variableXthat

is defined for all possible outcomes inS(so that a real number – not necessarily

unique – is assigned to each possible outcome) is called arandom variable(RV).

The outcome of the experiment may already be a real number and hence a random

variable, e.g. the number of heads obtained in 10 throws of a coin, or the sum of

the values if two dice are thrown. However, more arbitrary assignments are possi-

ble, e.g. the assignment of a ‘quality’ rating to each successive item produced by a

manufacturing process. Furthermore, assuming that a probability can be assigned

to all possible outcomes in a sample spaceS, it is possible to assign aprobability

distributionto any random variable. Random variables may be divided into two

classes, discrete and continuous, and we now examine each of these in turn.

30.4.1 Discrete random variables

A random variableXthat takes only discrete valuesx 1 ,x 2 ,...,xn, with proba-

bilitiesp 1 ,p 2 ,...,pn, is called a discrete random variable. The number of values

nfor whichXhas a non-zero probability is finite or at most countably infinite.

As mentioned above, an example of a discrete random variable is the number of

heads obtained in 10 throws of a coin. IfXis a discrete random variable, we can

define aprobability function(PF)f(x) that assigns probabilities to all the distinct

values thatXcan take, such that

f(x)=Pr(X=x)=

{

pi ifx=xi,

0otherwise.

(30.38)

A typical PF (see figure 30.6) thus consists of spikes, atvalid valuesofX, whose

height atxcorresponds to the probability thatX=x. Since the probabilities

must sum to unity, we require

∑n

i=1

f(xi)=1. (30.39)

We may also define thecumulative probability function(CPF) ofX,F(x), whose

value gives the probability thatX≤x,sothat

F(x)=Pr(X≤x)=

∑

xi≤x

f(xi). (30.40)