Fundamentals of Probability and Statistics for Engineers

Our derivation of Equation 4.81 has been purely analytical. In his theory of
Brownian motion, Einstein also obtained this result with

where R is the universal gas constant, T is the absolute temperature, N is
Avogadro’s number, and f is the coefficient of fr iction which, for liquid or
gas at ordinary pressure, can be expressed in terms of its viscosity and particle
size. Perrin, a F rench physicist, was awarded the N obel Prize in 1926 for his
success in determining, from experiment, Avogadro’s number.

4.5.3 Joint Characteristic Functions

The concept of characteristic functions also finds usefulness in the case of two
or more random variables. The development below is concerned with contin-
uous random variables only, but the principal results are equally valid in the
case of discrete random variables. We also eliminate a bulk of the derivations
involved since they follow closely those developed for the single-random-
variable case.
The joint characteristic function of two random variables X and Y,
is defined by

where t and s are two arbitrary real variables. This function always exists and
some of its properties are noted below that are similar to those noted for
Equations (4.48) corresponding to the single-random-variable case:

Furthermore, it is easy to verify that joint characteristic function
related to marginal characteristic functions

108 Fundamentals of Probability and Statistics for Engineers

Dˆ

2 RT

Nf

;… 4 : 82 †

XYt,s),

XY…t;s†ˆEfej…tX‡sY†gˆ

Z 1

1

Z 1

1

ej…tx‡sy†fXY…x;y†dxdy: … 4 : 83 †

XY… 0 ; 0 †ˆ 1 ;

XY…t;s†ˆXY…t;s†;

jXY…t;s†j 1 :

9

>>

=

>>

;

… 4 : 84 †

XYt,s)is

Xt) andYs)by

X…t†ˆXY…t; 0 †;

Y…s†ˆXY… 0 ;s†:

)

… 4 : 85 †