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:
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