Physical Chemistry Third Edition

9.1 Macroscopic and Microscopic States of Macroscopic Systems 385

the total number of members of the set equal towi. The mean can now be written as a

sum over the distinct members of the set

〈w〉

1

N

∑M

i 1

Niwi (9.1-2)

There are fewer terms in this sum than in the sum of Eq. (9.1-1) unless everyNiequals

unity, but the mean value is the same. We define

pi

Ni

N

(9.1-3)

The quantitypiis the fraction of the members of the set that are equal towi. It is also

equal to the probability that a randomly chosen member of the set will be equal towi.

The set ofpivalues is called aprobability distribution. We can now write

〈w〉

∑M

i 1

piwi (9.1-4)

From the definition ofpiin Eq. (9.1-3) these probabilities arenormalized, which

means that they sum to unity.

∑M

i 1

pi

1

N

∑M

i 1

Ni

N

N

1 (normalization) (9.1-5)

Exercise 9.1

A quiz was given to a class of 50 students. The scores were as follows:

Score Number of students

100 5

90 8

80 16

70 17

60 3

50 1

Find the mean score on the quiz without taking a sum of 50 terms.

We can also calculate the mean of a function of our values. Ifh(w) is some function

ofw, its mean value is

〈h〉

∑M

i 1

pih(wi) (9.1-6)