Page 127 - Fundamentals of Probability and Statistics for Engineers

110 Fundamentals of Probability and Statistics for Engineers

They are, respectively, the position of the particle after 2n steps and its position
after 3n steps relative to where it was after n steps. We wish to determine the
joint probability density function (jpdf) f XY (x, y) of random variables
X 0
X ˆ
n 1=2

and
Y 0
Y ˆ
n 1=2

for large values of n.
For the simple case of p ˆ q ˆ 1 , the characteristic function of each X k is [see
2
Equation (4.74)]
1 jt jt
t†ˆ Efe jtX k gˆ e ‡ e †ˆ cos t; 4:91†
2

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

0 0
tX sY
XY t; s†ˆ E exp‰ j tX ‡ sY†Šg ˆ E exp j ‡
f
n 1=2 n 1=2
* " #+)
n 2n 3n
j X X X
ˆ E exp t X k ‡ t ‡ s† X k ‡ s X k
n 1=2
kˆ1 kˆn‡1 kˆ2n‡1
t s ‡ t x
n h i o n
ˆ ;
n 1=2 n 1=2 n 1=2
4:92†

where (t) is given by Equation (4.91). The last expression in Equation (4.92) is
obtained based on the fact that the X k , k ˆ 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 f (x, y) from Equation (4.92) by using
XY
the inverse formula given by Equation (4.87). First, however, some simplifica-
tions are in order. As n becomes large,
t n t
h i n
ˆ cos
n 1=2 n 1=2
n
t t
2 4
ˆ 1 ‡ ... 4:93†
2
n2! n 4!
2
 e t =2 :

TLFeBOOK