Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

the series is absolutely convergent or that the integral exists, as the case may be.

From its definition it is straightforward to show that the expectation value has

the following properties:

(i) ifais a constant thenE[a]=a;

(ii) ifais a constant thenE[ag(X)] =aE[g(X)];

(iii) ifg(X)=s(X)+t(X)thenE[g(X)] =E[s(X)] +E[t(X)].

It should be noted that the expectation value is not a function ofXbut is

instead a number that depends on the form of the probability density function

f(x) and the functiong(x). Most of the standard quantities used to characterise

f(x) are simply the expectation values of various functions of the random variable

X. We now consider these standard quantities.

30.5.1 Mean

The property most commonly used to characterise a probability distribution is

itsmean, which is defined simply as the expectation valueE[X] of the variableX

itself. Thus, the mean is given by

E[X]=

{∑

∫ixif(xi) for a discrete distribution,

xf(x)dx for a continuous distribution.

(30.46)

The alternative notationsμand〈x〉are also commonly used to denote the mean.

If in (30.46) the series is not absolutely convergent, or the integral does not exist,

we say that the distribution does not have a mean, but this is very rare in physical

applications.

The probability of finding a1selectron in a hydrogen atom in a given infinitesimal volume

dVisψ∗ψdV, where the quantum mechanical wavefunctionψis given by

ψ=Ae−r/a^0.

Find the value of the real constantAand thereby deduce the mean distance of the electron

from the origin.

Let us consider the random variableR= ‘distance of the electron from the origin’. Since
the 1s orbital has noθ-orφ-dependence (it is spherically symmetric), we may consider
the infinitesimal volume elementdVas the spherical shell with inner radiusrand outer
radiusr+dr. Thus,dV=4πr^2 drand the PDF ofRis simply

Pr(r<R≤r+dr)≡f(r)dr=4πr^2 A^2 e−^2 r/a^0 dr.

The value ofAis found by requiring the total probability (i.e. the probability that the
electron issomewhere) to be unity. SinceRmust lie between zero and infinity, we require
that

A^2

∫∞

0

e−^2 r/a^04 πr^2 dr=1.