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2.160 Identification, Estimation, and Learning
Lecture Notes No. 4
February 17, 2006
3. Random Variables and Random Processes
Deterministic System:
Input
Output
In realty, the observed output is noisy and does not fit the model perfectly. In the
deterministic approach we treat such discrepancies as “error’.
Random Process:
An alternative approach is to explicitly model the process describing how the
“error” is generated.
Exogenous
Input
Process
Observed
Output
Process
Measurement
Noise
Noise
Random Process
Objective of modeling random processes:
•
Use stochastic properties of the process for better estimating parameters and
state of the process, and
•
Better understand, analyze, and evaluate performance of the estimator.
3.1 Random Variables
You may have already learned probability and stochastic processes in some subjects. The
following are fundamentals that will be used regularly for the rest of this course. Check if
you feel comfortable with each of the following definitions and terminology. (Check the
box of each item below.) If not, consult standard textbooks on the subject. See the
references at the end of this chapter.
1
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View Full Document 1) Random Variable
X
: Random variable is a function that maps every point in the sample space
to the real axis line.
F
2) Probability distribution
F
x
(x)
and probability density function
f
x
(x)
; PDF
X
(
x
)
=
Prob(
xX
)
≤
d
(
f
(
x
)
x
f
(
x
)
=
xF
)
x
x
dx
x
In the statistics and probability literature, the convention is that capital
X
represents
a random variable while lowercase
x
is used for an instantiation/realization of the random
variable.
3) Joint probability densities
Let
X
and
Y
be two random variables
f
XY
(x,y)
= Prob (X = x and Y = y, simultaneously)
4) Statistically independent (or simply Independent) random variables
f
XY
(x,y)
=
f
X
(x) f
Y
(y)
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.
 Spring '06
 HarryAsada

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