The Chemistry Maths Book, Second Edition

614 Chapter 21Probability and statistics

with mean (expectation value)

(21.37)

and standard deviation given by

(21.38)

An example of a continuous distribution, and probably the most important in statistics,

is the Gaussian distribution (normal distribution) discussed in Section 21.8.

EXAMPLE 21.13The uniform distribution

The simplest continuous distribution has probability

The probability density is constant throughout the range,

and the probability that the variable xlies in interval

x

1

1 < 1 x 1 < 1 x

2

is proportional to the width of the interval:

0 Exercise 23

EXAMPLE 21.14Radial distribution functions

A continuous distribution in three variables has probability density function such

that

is the probability that the variables lie in the given intervals; that is,(x, y, z)is a point

in the rectangular box whose sides arex

2

1 − 1 x

1

, y

2

1 − 1 y

1

, z

2

1 − 1 z

1

. The density function

is a function of position in the space of the variables, andρ(r)dvis the probability

that the point is in volume dvat position r. Some continuous three-dimensional

distributions of importance in chemistry are the electron probability distributions

obtained from solutions of the Schrödinger equation.

Px x x y y y z z z

z

z

y

y

x

x

()

121 212

1

2

1

2

1

2

<< ; << ; << =ZZZρ(()x y z dxdydz,,

Px x x x dx

xx

ba

x

x

()()

12

21

1

2

<< = =

−

−

Z ρ

ρ()x

ba

axb

=

−

<<











1

0

if

otherwise

σμρ

22

=−Z

a

b

()()xxdx

μρ=〈 〉=xxxdx

a

b

Z ()

.

..

...

..

...

..

...

..

...

.

...

..

...

..

...

..

...

.........

........

....

....

........

.......

x

ρ(x)

0 ab

b−a

...........................................................................................................................................................................................................

Figure 21.6