SUMMARY AND EXAMPLES
Preparing final exam: (Dec. 15, W. 9:45 - 11:45 am).
Importance of materials:
1. All homework solutions;
2. Lecture notes examples and past exam questions;
3. Relevant textbook content.
EE/STAT 322, #25 1
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Study Guide for Final Exam Preparation | STAT 322, Exams of Statistics
Material Type: Exam; Class: PROBAB MTH,ELEC ENG; Subject: STATISTICS; University: Iowa State University; Term: Unknown 1989;
Typology: Exams
Pre 2010
1 / 19
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SUMMARY AND EXAMPLES
- Relevant textbook content.2. Lecture notes examples and past exam questions;1. All homework solutions;Importance of materials: Preparing final exam: (Dec. 15, W. 9:45 - 11:45 am). EE/STAT 322, #
COVERED TOPICS
Outline:
Set Theory, The Axiomatic Approach.Random Experiments and Events, Definition of Probability, ElementaryIntroduction To Probability:
Set Operations, Conditional Probability, Independence
Combined Experiment Bernoulli Trials and its Applications
Concept of A Random Variable:
Distribution Function (CDF), Probability Density Function (PDF),
Mean Values and Moments
EE/STAT 322, #
COVERED TOPICS (CONT.)
Characteristic Function; Moment generating function (MGF).
Definition of transform
Using transform to evaluate moments
MGF of Sum of Independent RVs
Elements of Statistics:
Sample Mean, Sample Variance – Biased and Unbiased.
PDFs of Samples and Sample mean,
Confidence Intervals and Hypothesis Testing
and Nonstationary; Ergodic and Nonergodic.Continuous and Discrete; Deterministic and Nondeterministic; StationaryRandom Processes
EE/STAT 322, #
COVERED TOPICS (CONT.)
Correlation Functions
Auto-correlation Functions – Properties, examples,
Crosscorrelation Functions
Relation to stationary, ergodic processes.
Correlation matrix, covariance matrix.
White NoiseRelation of PSD to Fourier Transform, Mean-Square values from PSD,Power Spectral Density (PSD)
Central Limit Theorem (CLT)
Facts about Gaussian PDF and MGF
Markov and Chebyshev inequalites.
Linear MMSE estimation, Orthogonality Principle, MSE
EE/STAT 322, #
EXAMPLE: MMSE ESTIMATE (CONT.)
(2) Method I:
J
min
E
[
Y
aX
2 ] =
E
[
Y
2 ] +
E
[
a 2 X 2 ] − E
[
aXY
]
σ Y 2
ρ 2 σ Y 2
σ X 2
σ X 2
ρσ
Y
σ X
ρσ
Y
σ X
σ Y 2
ρ 2 σ Y 2
ρ 2 σ Y 2
ρ 2 ) σ Y 2
.
J Method II: (orthogonal principle:) min
= E [ | Y −
aX
2 ] =
E
[
Y 2 ] − E [ a 2 X 2 ]
σ Y 2
ρ 2 σ Y 2
σ X 2
σ X 2
ρ 2 ) σ Y 2
.
E
[ ˆ
Y
Y
Yˆ
)] =
E
[
aX
Y
aX
)] =
aE
[
XY
] − a 2 E [ X 2 ]
ρσ
Y
σ X
ρσ
X
(^) σ
Y − ( ρ 2 σ
Y 2
σ X 2
σ X 2
ρ 2 σ Y 2
ρ 2 σ Y 2
EE/STAT 322, #
EXAMPLE: MGF
Let
X
be a random variable with the following pmf,
P
X
P
X
, and
P
X
Find the MGF of
X
, and use it to evaluate
E [ X ] , E [
X
2 ] , and
E [ X 3 ].
Solution:
M
s ) =
E
[
e sX
] =^
2 1 e s
4 1 e 2 s
4 1 e 3 s .
E
[
X
] =
d ds
M
s ) | s =
E
[
X
2 ] =
d 2
ds
2 (^) M
s ) | s =
E
[
X
3 ] =
d 3
ds
3 (^) M
s ) | s =
EE/STAT 322, #
EXAMPLE: MGF (CONT.)
Y
and
Z
are independent RVs, where
Y
is exponentially distributed with
parameter 2, and
Z
is Gaussian with mean 3 and variance 4.
(a) Find the MGF for
Z
; (b) Find the MGF for
Y
Z
Solution:
f Y
( y ) = 2
e −
2 y , and
Z
N
M
Y
( s ) = 2
s ) , and
M
Z
( s ) =
e μs
σ 2 s 2 / 2 = e 3 s
s 2 .
(a)
M
2 Z
s ) =
E
[
e s (
Z +3)
] =
e 3 s E [ e s 2 Z
] =
e 3 s M
Z (
s )
e 3 s e 3(
s )+2(
s ) 2 = e 9 s
s 2 .
(b)
M Y + Z ( s
M Y ( s ) M Z ( s
2
2 − s e 3 s
s 2 .
EE/STAT 322, #
EXAMPLE (GAUSSIAN CONDITIONAL
DISTRIBUTION)
Let
X
be a Gaussian RV with mean 4 and variance 16.
(a) Find the probability
P
X <
and
P
X >
X Solution:
σ x
= 4
P
X <
Q
Q
P
X >
Q
Q
EE/STAT 322, #
(c) Using Chebyshev inequality, find the bound for
P
X
Compute
P
X
again using Gaussian-Q function.
Solution:
P
X
P
X
X
σ
2
= 16
, and
C
, Chebyshev inequality leads to
P
X
X
> C
σ 2 /C
2
⇒
P
| X − 4 | >
So
P
X
P
| X − 4 | >
P
X
Q
[
− Q ( − 2.
5)]
Q
Note
1 − Q ( − 2.
Q
EE/STAT 322, #
13
EXAMPLE: CORRELATION AND COVARIANCE
Two Gaussian RVs
X
and
Y
have means of 1 and 2, respectively.
They
have variances of 1 and 9, respectively.
Their correlation coefficient is
ρ
(a) Find the variance of their sum
X
Y
σ x 2
y
respectively.
(b) Find
E
XY
(a) Solution:
σ x 2
y
=
σ x 2
σ y 2
- 2
ρσ
x σ y
(b)
E
[
XY
] =
X
Y¯
ρσ
x σ y
= 1
EE/STAT 322, #
EXAMPLE (PSD)
Use
the
Parsevel’s
theorem
to
evaluate
the
following
integral:
∞ −∞
sin(
ω )
4 ω
1
ω 2
dω
F Solution:
(^) ( ω
) =
sin(
ω )
4 ω
= (sinc(
T f
f (^) ( t ) =
F
(^) −
1 { F
(^) ( ω
) }
=
8 1 ,
t | ≤
g ( t ) =
F − 1 { G ( ω ) } = F − 1 { 1
ω 2
4 1 e −
2 | t | .
Using Parseval’s theorem:
∞ −∞
f (^) ( t ) g ( t )
dt
1
2 π
∫
∞ −∞
F
ω ) G ( − ω )
dω.
∞
−∞
sin(
ω )
4 ω
1
ω 2
dω
π
∫
∞ −∞
f (^) ( t ) g ( t )
dt
π
4 −
4
8 1 4 1 e −
2 | t | dt
2 π
32
4 0 2 e − 2 t
dt
(^16) π
e −
2 t | 40
=
(^16) π
− e − 8 ).
EE/STAT 322, #
EXAMPLE (PSD)
A stationary process
X
t )
has the PSD
S
X
(^) ( ω
) =
1
ω 2
π 4 δ ( ω ).
(a) Find
R
X
(^) ( τ (^) )
. (b) Find
X
X
2
and
σ X 2
(^).
Solution:
(a)
R
X
(^) ( τ (^) ) =
F − 1 { S X
ω
) } .
F
−
1 {
2 Aβ
ω 2
β 2
- 2
π 4 δ ( ω ) } =
Ae
−
β | τ (^) |
- 4
, where
A
2 1 ,
β
R
X
(^) ( τ (^) ) =
2 1 e −|
τ (^) |
- 4
(b) Since
X
t )
is stationary, we get
R
X
(^) ( ∞
X
2 ,
⇒
X
2
=
2 1 e −∞
X
X
2
=
R
X
(^) (0) =
2 1
- 4 = 4
σ X 2
X
2
−
X
2
= 4
EE/STAT 322, #
LESS IMPORTANT CONTENT
#6, Power PDF, Maxwell PDF, Lognormal PDF, Chi PDF.
#7, p. 7-9, Delta PDF, p. 14 (example 2-7.1)
#11, p.11 (residue method for inverse Laplace transform).
processes (several examples).#17, #18, #19 (include P. 8, Ex 6-8.2) - correlation function of binary
#18, p.13, autocorrelation function of time derivation process.
Law
of
large
numbers
(except
the
inequalities);
types
of
convergence.
Nonlinear
MMSE
estimation;
estimation
based
on
several
measurement.
EE/STAT 322, #