# Connexions

You are here: Home » Content » Examples of Stochastic Processes

### Recently Viewed

This feature requires Javascript to be enabled.

# Examples of Stochastic Processes

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 =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 : ωR ω is a random variable defined over π π and is assumed to be a uniform random variable; i.e., f 0 θ={12π  if   π π 0  otherwise   f 0 θ 1 2 0

F X t b=Pr X t b=Prcos2π f 0 t+Θb=Prπ2π f 0 t+Θarccosb+Prarccosb2π f 0 t+Θπ 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 t12πd θ +arccosb2π f 0 tπ2π f 0 t12πd θ =(2π2arccosb)12π 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 11πarccosx={1π1x2  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.
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)
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 nTt<(n+1)T n T t n 1 T . If tails, then X t =-1 X t -1 for nTt<(n+1)T n T t n 1 T .

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 nT t 1 <(n+1)T n T t 1 n 1 T and nT 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 nT t 1 <(n+1)T n T t 1 n 1 T and mT t 2 <(m+1)T m T t 2 m 1 T with nm n m
P p X t 2 X t 1 x 2 x 1 ={0  if  ( x 2 x 1 )(nT t 1 )( t 2 <(n+1)T) p X t 1 x 1   if  ( x 2 = x 1 )(nT 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)

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks