CPE 332 Computer Engineering Mathematics II

1
CPE 332Computer Engineering Mathematics II

• Part II, Chapter 4
• Probability and Random Variable
• (Review)

2
Today Topics

• HW 3 Due
• Probability
• Random Variable
• PDF
• Expectation
• Concept of Random Process

3
Definition

• Outcome/Sample Point
• ???????????????????????? ????????????????
• Sample Space
• Set ?????????????????
• Event
• ???????????????????
• ????? Event A, ???????? N ?????
  ?????????????????????????????? A NA ?????

4
Example Dice Roll

• Sample Space 1,2,3,4,5,6
• P(1)P(2) P(6)1/6
• Laplace Definition of Probability
• ???????? Member ?? Sample Space
  ???????????????????
• A Even
• A 2,4,6
• P(A) 2,4,6/S 3/6 ½

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Mutually Exclusive

• A ME B
• ????????? Event A ?????? Event B ??????
• ??????????????????????? ??????????????
• ?????????? A Even, B Odd
• A ME B
• P(AB) or P(A union B) P(A) P(B)
• ????????????????????? Vein Diagram

A
B
B
A
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Mutually Exclusive

• ME
• A?B ?
• A?BAB
• P(A?B)P(A)P(B)

B
A

• Non ME
• A?B ??
• A?BAB- A?B
• Inclusion-Exclusion Principle

B
A
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3 AXIOMS OF Probability

• 1. P(S) 1
• ???????????? ????????????????? Sample Space
• 2. 0 lt P(A) lt 1
• ?????? Probability ??????????????? 0 ??? 1
• 3. ME P(AB)P(A)P(B)
• Mutually Exclusive Probability ??? Union ???
  Event ??????????????? Probability ???????? Event

8
Conditional Probability

• Probability ??? Event ????? ????????????? ???
  Event ????????????????
• Probability ????????????? Event ?????????????
• Probability ?????????????????? Event ????????????
• ??????????????? Statistical Independent
• ?????????

9
Bayes Rule
E1
E3
E6
A
E9
E4
E2
E7
E8
E5
10
Properties
11
Example 1

• ????????????????? Digital ???? Frame ???? 50 ???
  ???????????????????????????????? Frame
  ????????????????? ?????????? Error
  ???????????????????(BER Bit Error Rate)
  ?????????????????????????? 1/1000 ??????? Frame
  ?????????? Error ???????????????????????? (FER
  Frame Error Rate) ?????????? Error ????? Bit ????
  Independent

12
Example 2

• ???????? ?????????????????????????????????????
  CPE332 ???????????? 0.5 ??????????????????????????
  ?????????????????? 0.4 ??????? CPE332
  ???????????????????? 40 ????????????????? 25 ??
  ????????????????????????????????????

• ???????????????????????????????
• ??? Sample Space ?????????????????????(M)
  ???????????????(F) ??????? S M,F
• ?????????? Set ??????????? ME ??? Sample Space
  ??? Partition ??????? Partition
• PM40/65 ??? PF25/65
• ??? Event A ?????????????????????????????????
  ??????
• PA\M0.5 ??? PA\F0.4
• ??????????? Partition

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Example3

• ????????????????? 2 ?????????????????????????????
  ??????????????? CPE332 ????????????????
  ???????????????????????????????????????
  ???????????????????????????????? ??????????
• 1.????????????????????????????????????????????????
  ?
• 2. Probability ???????????????????????????????????
  ?????

• 1. ????????? PF\A

• 2. ????????? PFAPF?A

14
Random Variables

• ????????????????????????????????? Sample Point ??
  Sample Space ????????? Variable
  ??????????????????????????? ??????????????????????
  ??????????????? ??????????? Variable ???????????
  Random Variable ??????????????????????? Random
  Variable ?????????? Capital
• ??????????? ???????? RV ??????????????????????????
  ??????????????????????????????
• Mean
• Variance
• Etc.

15
Random Variables

• ???????????????????????(Sample Space)??? Infinite
  Set ?????????????????????????????????????????
• ?????? Continuous Random Variable
• ???????????????? Finite Set ????????????? Set
  ????????? ????????? Integer
• ?????? Discrete Random Variable
• Probability ????????????????????????????? ???
  Probability ??? Random Variable
  ?????????????????????
• ????????????????????????? RV ?????? Plot ???
  Probability(y-axis) ????????? Random
  Variable(x-axis)
• Cumulative Distribution Function (CDF)
• Probability Density Function (PDF)

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CDFCumulative Distribution Function of RV X
FX(x)
1.0
FX (10) PX 10
x
x 10
CDF of Normal (Gaussian) Distribution Continuous
17
CDF Properties
18
CDF ???????????? Discrete RV

• ??? ??? 0 ??? ???? 1

FX(x) P(X x)
1.0
0.5
0
x
1
0
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CDF ????????????? Discrete RV

• ??? ??????????????????????????

FX(x) P(X x)
1.0
0.5
0
x
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PDF Probability Density Function
f(x)
Area ?f(x)dx 1
x
PDF of Normal(Gaussian) Distribution Continuous
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Properties of PDF
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Discrete Version

• ?????? Variable ????????????
• RV X ????????????? Xxi
• F(x) P(Xx)
• Function ????????????????????????????
• ??????????????????? Domain ??? x
• Function ???????????????????
• Monotonic Increasing Function ??? 0 ??? 1
• f(xi) P(X xi)
• ????????????? ???????????? ???????????????????????
  
• ?????????? Probability Mass Function
• ?f(xi)1 ????

23
f(x)
PMF
x
F(x)
CDF
x
24
Statistical Average
25
Importance Expectation
26

• Note
• ????????? Data ????????????????????????
  ?????????????? Estimate ??? Mean ??? Mean Square
  ???????? ???

27
Joint Moment
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Correlation

• G(x,y) XY

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Covarience
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Correlation and Covarience
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PDF ????????
PXx
PXltx
1.0
q
p
x
1
0
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Binomial Distribution
b(k10,0.2)
b(k10,0.5)
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Geometric Distribution
P0.5
P0.2
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Uniform Distribution
f(x)
1/(b-a)
x
a
b
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Gaussian Distribution
36
Gaussian
Area 1
37
Jointly Gaussian X, Y
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P 0
Volumn 1
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P 0.5
40
P 0.9
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P 0.95
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P -0.99
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Exponential Distribution
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Exponential Distribution
A 1
Area 1
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Poisson Distribution
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Lambda 1
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Lambda 3
48
Lambda 5
49
Lambda 8
50
Lambda 12
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Lambda 18
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Random Process

• ????? Random Variable ???? Function ???????
• ????????????????? Random ???? Noise
• ?????? Packet ?? Network
• ???????????????????
• ?????????????????????????????
• ???????? ????????????????? ???????????????????????
  Random ????? PDF ????????????????
• ?????????????? PDF ????????????
  ??????????????????????????
• Random Variable ??????? Function (Random Function
  ???? Analytic Function) ??????? ???????? Random
  Process ???? Stochastic Process
• ??? RV X ?????? X(t)
• Mean ??? Variance ?????? Function ???????????
• ????????????????????????? RV ?????????????????????
  ???????????????

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Homework IV Due Next Week

• Download ??? Web
• Next Week Chapter V
• Random Process
• MarKov Process and MarKov Chain