Modern Thermodynamics

- Chapter 1

44

Appendix 1.2 Elementary Concepts in Probability Theory

In the absence of a deterministic theory that enables us to calculate the quantities of interest to us, one

uses probability theory. Let xk in which k=1,2,,3, ….n represent all possible n values of a random

variable x. For example, x could be the number molecules at any instant in a small volume of 1 nm3

within a gas or the number of visitors at a web site at any instant of time. Let the corresponding

probabilities for these n values of x be P(xk). Since xk, k=1,2, …n, represent all possible states:

P(xk ) =1

k=1

n

∑

(A1.2.1)

Average values

We shall denote the average value of a quantity A, by A. Thus, the average value of x would be

x = xkP(xk )

k=1

n

∑

(A1.2.2)

Similarly, the average value of x2 would be:

x2

= xkP(xk

2

)

k=1

n

∑

(A1.2.3)

More generally, if f(xk) is a function of x, its average value, would be:

f = )P(xk ).

k=1

n

∑f(xk

If the variable x takes continuous values in the range (a, b), then the average values are written as

integrals:

x = xP(x)dx

a

b

∫

f = f(x)P(x)dx

a

b

∫

(A1.2.4)

For a given probability distribution, the standard deviation, s, is defined as:

s=

− x

(x )2

(A1.2.5)

Some common probability distributions:

Binomial Distribution: This is the probability distribution associated with two outcomes H and T (such

as a coin toss) with probabilities p and (1-p). The probability that in N trials, m are H and (N–m) are T is

given by: