Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.5 PROPERTIES OF DISTRIBUTIONS

Integrating by parts we findA=1/(πa^30 )^1 /^2. Now, using the definition of the mean (30.46),
we find

E[R]=

∫∞

0

rf(r)dr=

4

a^30

∫∞

0

r^3 e−^2 r/a^0 dr.

The integral on the RHS may be integrated by parts and takes the value 3a^40 /8; conse-
quently we find thatE[R]=3a 0 /2.

30.5.2 Mode and median

Although the mean discussed in the last section is the most common measure

of the ‘average’ of a distribution, two other measures, which do not rely on the

concept of expectation values, are frequently encountered.

Themodeof a distribution is the value of the random variableXat which the

probability (density) functionf(x) has its greatest value. If there is more than one

value ofXfor which this is true then each value may equally be called the mode

of the distribution.

ThemedianMof a distribution is the value of the random variableXat which

the cumulative probability functionF(x) takes the value^12 ,i.e.F(M)=^12. Related

to the median are the lower and upper quartilesQlandQuof the PDF, which

are defined such that

F(Ql)=^14 ,F(Qu)=^34.

Thus the median and lower and upper quartiles divide the PDF into four regions

each containing one quarter of the probability. Smaller subdivisions are also

possible, e.g. thenth percentile,Pn, of a PDF is defined byF(Pn)=n/100.

Find the mode of the PDF for the distance from the origin of the electron whose wave-

function was given in the previous example.

We found in the previous example that the PDF for the electron’s distance from the origin
was given by

f(r)=

4 r^2

a^30

e−^2 r/a^0. (30.47)

Differentiatingf(r) with respect tor,weobtain

df

dr

=

8 r

a^30

(

1 −

r

a 0

)

e−^2 r/a^0.

Thusf(r) has turning points atr=0andr=a 0 ,wheredf/dr= 0. It is straightforward
to show thatr= 0 is a minimum andr=a 0 is a maximum. Moreover, it is also clear that
r=a 0 is a global maximum (as opposed to just a local one). Thus the mode off(r) occurs
atr=a 0 .