Fundamentals of Probability and Statistics for Engineers

They are, respectively, the position of the particle after 2 n steps and its position
after 3 n steps relative to where it was after n st eps. We wish to determine the
joint probability density function (jpdf) fXY (x,y) of random variables

and

for large values of n.
For the simple case of p q^12 , the characteristic function of each Xk is [see
Equation (4.74)]

and, following Equation (4.83), the joint characteristic function of X and Y is

where (t) is given by Equation (4.91). The last expression in Equation (4.92) is
obtained based on the fact that the Xk,k 1,2,...,3n, are mutually independ-
ent. It should be clear that X and Y are not independent, however.
We are now in the position to obtain fXY (x,y) from Equation (4.92) by using
the inverse formula given by Equation (4.87). F irst, however, some simplifica-
tions are in order. As n becomes large,

110 Fundamentals of Probability and Statistics for Engineers

Xˆ

X^0

n^1 =^2

Yˆ

Y^0

n^1 =^2

ˆ ˆ

…t†ˆEfejtXkgˆ

1

2

…ejt‡ejt†ˆcost; … 4 : 91 †

XY…t;s†ˆEfgexp‰j…tX‡sY†Š ˆE exp j

tX^0

n^1 =^2

‡

sY^0

n^1 =^2



ˆE exp

j

n^1 =^2



t

Xn

kˆ 1

Xk‡…t‡s†

X^2 n

kˆn‡ 1

Xk‡s

X^3 n

kˆ 2 n‡ 1

Xk

)*+"

ˆ 

t

n^1 =^2





s‡t

n^1 =^2

hi



x

n^1 =^2

non

;

… 4 : 92 †



ˆ



t

n^1 =^2

hin

ˆcosn

t

n^1 =^2



ˆ 1 

t^2

n 2!

‡

t^4

n^24!

...

n

et

(^2) = 2

… 4 : 93 †