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Examples of Stochastic Processes

Module by: Roy Ha, Dinesh Rajan, Mohammad Borran, Behnaam Aazhang. E-mail the authors

Summary: Examples of Stochastic processes.

This modules contains examples of stochastic processes. For the material related to these examples, please refer to Introduction to Stochastic Processes.

Example 1

X t =cos2⁢π⁢ f 0 ⁢t+Θ⁢ω X t 2 f 0 t Θ ω where f 0 f 0 is the deterministic carrier frequency and Θ⁢ω Θ ω : ω∈R ω is a random variable defined over −π π and is assumed to be a uniform random variable; i.e., f 0 ⁢θ={12⁢π  if   −π π 0  otherwise   f 0 θ 1 2 0

F X t ⁢b=Pr X t ≤b=Prcos2⁢π⁢ f 0 ⁢t+Θ≤b=Pr−π≤2⁢π⁢ f 0 ⁢t+Θ≤−arccosb+Prarccosb≤2⁢π⁢ f 0 ⁢t+Θ≤π F X t b X t b 2 f 0 t Θ b 2 f 0 t Θ b b 2 f 0 t Θ
(1)
∫(−π)−2⁢π⁢ f 0 ⁢t(−arccosb)−2⁢π⁢ f 0 ⁢t12⁢πd θ +∫arccosb−2⁢π⁢ f 0 ⁢tπ−2⁢π⁢ f 0 ⁢t12⁢πd θ =(2⁢π−2⁢arccosb)⁢12⁢π θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2
(2)
f X t ⁢x=dd x 1−1π⁢arccosx={1π⁢1−x2  if  |x|≤10  otherwise   f X t x x 1 1 x 1 1 x 2 x 1 0
(3)
This process is stationary of order 1.
Figure 1
Figure 1 (Figure3-3a.png)
X t =cos2⁢π⁢ f 0 ⁢t+Θ X t 2 f 0 t Θ
(4)
Pr X t 2 ≤ b 2 | X t 1 = x 1 =???? X t 1 x 1 X t 2 b 2 ????
(5)
( X t 1 = x 1 =cos2⁢π⁢ f 0 ⁢t+Θ)⇒(Θ=arccos x 1 −2⁢π⁢ f 0 ⁢t) X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t
(6)
X t 2 =cos2⁢π⁢ f 0 ⁢ t 2 +arccos x 1 −2⁢π⁢ f 0 ⁢ t 1 =cos2⁢π⁢ f 0 ⁢( t 2 − t 1 )+arccos x 1 X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1
(7)
Figure 2
Figure 2 (Figure3-3b.png)
F X t 2 , X t 1 ⁢ b 2 b 1 =∫−∞ b 1 Pr X t 2 ≤ b 2 | X t 1 = x 1 ⁢ f X t 1 ⁢ x 1 d x 1 F X t 2 , X t 1 b 2 b 1 x 1 b 1 X t 1 x 1 X t 2 b 2 f X t 1 x 1
(8)

Example 2

Every TT seconds, a fair coin is tossed. If heads, then X t =1 X t 1 for n⁢T≤t<(n+1)⁢T n T t n 1 T . If tails, then X t =-1 X t -1 for n⁢T≤t<(n+1)⁢T n T t n 1 T .

Figure 3
Figure 3 (Figure3-4.png)
p X t ⁢x={12  if  x=112  if  x=−1 p X t x 1 2 x 1 1 2 x 1
(9)
X t X t is stationary of order 1.

Second order probability mass function

P X t 1 X t 2 x 1 x 2 = p X t 2 | X t 2 ⁢ x 2 | x 1 ⁢ p X t 1 ⁢ x 1 P X t 1 X t 2 x 1 x 2 p X t 2 | X t 2 x 2 | x 1 p X t 1 x 1
(10)

The conditional pmf

p X t 2 | X t 1 ⁢ x 2 | x 1 ={0  if   x 2 ≠ x 1 1  if   x 2 = x 1 p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1
(11)
when n⁢T≤ t 1 <(n+1)⁢T n T t 1 n 1 T and n⁢T≤ t 2 <(n+1)⁢T n T t 2 n 1 T for some nn.
p X t 2 | X t 1 ⁢ x 2 | x 1 = p X t 2 ⁢ x 2 p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2
(12)
for all x 1 x 1 and for all x 2 x 2 when n⁢T≤ t 1 <(n+1)⁢T n T t 1 n 1 T and m⁢T≤ t 2 <(m+1)⁢T m T t 2 m 1 T with n≠m n m
P p X t 2 X t 1 x 2 x 1 ={0  if  ( x 2 ≠ x 1 )∧(n⁢T≤ t 1 )∧( t 2 <(n+1)⁢T) p X t 1 ⁢ x 1   if  ( x 2 = x 1 )∧(n⁢T≤ t 1 )∧( t 2 <(n+1)⁢T) p X t 2 ⁢ x 2 ⁢ p X t 1 ⁢ x 1   otherwise   P p X t 2 X t 1 x 2 x 1 0 x 2 x 1 n T t 1 t 2 n 1 T p X t 1 x 1 x 2 x 1 n T t 1 t 2 n 1 T p X t 2 x 2 p X t 1 x 1
(13)

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