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Examples of Stochastic Processes

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

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Summary: Examples of Stochastic processes.

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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 =cos2π f 0 t+Θω X t 2 f 0 t Θ ω where f 0 f 0 is the deterministic carrier frequency and Θω Θ ω : ω∈ℝ ω is a random variable defined over -ππ and is assumed to be a uniform random variable; i.e., f 0 θ=12πif-ππ0otherwise f 0 θ 1 2 0

F X t b=Pr X t ≤b=Prcos2π f 0 t+Θ≤b=Pr-π≤2π f 0 t+Θ≤-arccosb+Prarccosb≤2π 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−2π f 0 t-arccosb−2π f 0 t12πdθ+∫arccosb−2π f 0 tπ−2π f 0 t12πdθ=2π−2arccosb12π 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=ddx1−1πarccosx=1π1−x2if|x|≤10otherwise 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 =cos2π 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 =cos2π f 0 t+Θ⇒Θ=arccos x 1 −2π f 0 t X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t (6)
X t 2 =cos2π f 0 t 2 +arccos x 1 −2π f 0 t 1 =cos2π 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 nT≤t<n+1T n T t n 1 T . If tails, then X t = -1 X t -1 for nT≤t<n+1T n T t n 1 T .

Figure 3
Figure 3 (Figure3-4.png)
p X t x=12ifx=112ifx=-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 =0if x 2 ≠ x 1 1if 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 nT≤ t 1 <n+1T n T t 1 n 1 T and nT≤ t 2 <n+1T 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 nT≤ t 1 <n+1T n T t 1 n 1 T and mT≤ t 2 <m+1T m T t 2 m 1 T with n≠m n m
P p X t 2 X t 1 x 2 x 1 =0if x 2 ≠ x 1 ∧nT≤ t 1 ∧ t 2 <n+1T p X t 1 x 1 if x 2 = x 1 ∧nT≤ t 1 ∧ t 2 <n+1T 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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