Fundamentals of Probability and Statistics for Engineers

The distribution given by Equation (4.72) is called a gamma distribution,
which will be discussed extensively in Section 7.4.

Example 4.17.In 1827, R obert Brown, an English botanist, noticed that small
particles of matter from plants undergo erratic movements when suspended in
fluids. It was soon discovered that the erratic motion was caused by impacts on
the particles by the molecules of the fluid in which they were suspended. This
phenomenon, which can also be observed in gases, is called Brownian motion.
The explanation of Brownian motion was one of the major successes of statistical
mechanics. In this example, we study Brownian motion in an elementary way by
using one-dimensional random walk as an adequate mathematical model.
Consider a particle taking steps on a straight line. It moves either one step to
the right with probability p, or one step to the left with probability
The steps are always of unit length, positive to the right and
negative to the left, and they are taken independently. We wish to determine the
probability mass function of its position after n steps.
Let Xi be the random variable associated with the ith st ep and define

Then random variable Y, defined by

gives the position of the particle after n steps. It is clear that Y takes integer
values between and n.
To determine pY (k), we first find its characteristic function. The
characteristic function of each Xi is

It then follows from Equation (4.71) that, in view of independence,

Let us rewrite it as

106 Fundamentals of Probability and Statistics for Engineers

qp‡qˆ1).

Xiˆ

1 ; if it is to the right;

 1 ; if it is to the left.



… 4 : 73 †

YˆX 1 ‡X 2 ‡‡Xn;

n

nkn,

Xi…t†ˆEfejtXigˆpejt‡qejt: … 4 : 74 †

Y…t†ˆX 1 …t†X 2 …t†...Xn…t†

ˆ…pejt‡qejt†n: … 4 : 75 †

Y…t†ˆejnt…pe2jt‡q†n

ˆ

Xn

iˆ 0

n

i



piqniej…^2 in†t