The Chemistry Maths Book, Second Edition

21.7 Continuous distributions 613

21.7 Continuous distributions

The distributions considered so far have been discrete, involving the discrete variable

x(or a variable that is treated as discrete for practical purposes). Each value of such

a variable has an observed relative frequency and a corresponding theoretical

probability. Such distributions describe processes that involve the counting of discrete

events, and one example is the binomial distribution described in Section 21.5.

Processes that involve the measurement of a continuous quantity are described by

continuous distributions, for which the variable xcan have any value in a continuous

range,a 1 ≤ 1 x 1 ≤ 1 bsay. The number of possible outcomes is then infinite, and the

probability of a particularoutcome is not defined (it is effectively zero). We define,

instead, the probability that the value of xlies in a specified intervalx

1

1 ≤ 1 x 1 ≤ 1 x

2

:

(21.33)

whereρ(x)is called the probability density distribution(or probability density

function, or just probability density). Whenx

2

1 − 1 x

1

1 = 1 ∆xis small enough, the probability

in the interval xtox 1 + 1 ∆xis

P(x 1 → 1 x 1 + 1 ∆x) 1 ≈ 1 ρ(x)∆x (21.34)

with equal sign for an infinitesimal intervaldx. The total probability (the normalization

ofρ(x))is

(21.35)

Figure 21.5 illustrates the graphical interpretation. The total area under the curve

between aand bis equal to 1 and represents the total probability (equation (21.35)).

The shaded region has area equal to the probability given by (21.33).

The properties of a continuous distribution are obtained by replacing the sums

for the discrete distribution by integrals over the range of possible values, and the

expectation values are then (see equations (21.11) to (21.23) for the discrete case)

(21.36) 〈〉=ffxxdx

a

b

Z ()()ρ

Pa x b x dx

a

b

()()≤≤ =Z ρ = 1

Px x x x dx

x

()()

12

1

2

≤≤ =Z ρ

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x

ρ(x)

x

1

x

2

0 a

b

P(x

1

≤x≤x

2

)

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Figure 21.5