•1
Lecture 15
Stochastic/Random
Processes
Basics
•
First recall an RV is a mapping from
which we denote by
or
•
When we add the notion of time to an RV to
obtain
we have a stochastic process.
•
It is a rule for assigning to every
a
function
R
S
→
R
S
X
→
:
)
(
ω
X
)
(
ξ
X
)
,
(
ξ
t
X
S
∈
ξ
).
,
(
ξ
t
X
•
It is a class of time functions depending on
the parameter
, or a function of t and
of
.
If
is a continuous time process.
If
or N, then
is a discrete time
process.
•
A stochastic process may be viewed in any of
the following ways:
a)
An ensemble of functions
where
are
variables.
b)
A sample time function of a given process
when
is fixed.
ξ
ξ
)
(
t
X
R
t
→
∈
Z
t
∈
)
(
t
X
)
,
(
ξ
t
X
ξ
,
t
ξ
c) If t is fixed, and
is variable, then it is a
RV.
d) If t and
are fixed, X(t) is a number.
ξ
ξ

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