Download Chapter 2 Discrete Random Variables (Part II)

Chapter 2
Discrete Random Variables (Part II)
Chih-Wei Tang
Visual Communications Lab
Department of Communication Engineering,
National Central University
Jhongli, Taiwan
2017 Spring
Outline
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Definitions
Families of Discrete Random Variables
Cumulative Distribution Function (CDF)
Averages
Derived Random Variable
Variance and Standard Deviation
Conditional Probability Mass Function
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1. Discrete random variable
2. Continuous random variable
Definition 2.4
3
Cumulative Distribution Function
FX ( x)  0, if x  xmin ,
FX ( x)  1, if x  xmax .
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Related Theorem of
Cumulative Distribution Function (1/2)
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Related Theorem of
Cumulative Distribution Function (2/2)
• Going from left to right on the x-axis, FX (x) starts at
zero and ends at one.
• The CDF never decreases as it goes from left to right.
• For a discrete random variable xi  S X, there is a
jump (discontinuity) at each value of X . The height
of the jump at xi is PX ( xi ) .
• Between jumps, the graph of the CDF of the
discrete random variable X is a horizontal line.
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Outline
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Definitions
Families of Discrete Random Variables
Cumulative Distribution Function (CDF)
Averages
Derived Random Variable
Variance and Standard Deviation
Conditional Probability Mass Function
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How to Describe A Collection of Numerical
Observations with A Single Number?
1
2
3
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Outline
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Definitions
Families of Discrete Random Variables
Cumulative Distribution Function (CDF)
Averages
Derived Random Variable
Variance and Standard Deviation
Conditional Probability Mass Function
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Example #1 – Derived Random Variable
Random variable
PX (x)
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Derived
Random variable
Y  g( X )
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Definition – Derived Random Variable
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Example #2 – Derived Random Variable (1/2)
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Example #2 – Derived Random Variable (2/2)
v  3,2,1, 0, 1, 2, 3.
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1
1 9
y  , 2, , 0, ,2, .
2
2
2 2
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Expected Value of
A Derived Random Variable
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Outline
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Definitions
Families of Discrete Random Variables
Cumulative Distribution Function (CDF)
Averages
Derived Random Variable
Variance and Standard Deviation
Conditional Probability Mass Function
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Definition – Variance
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Example – Variance (1/2)
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Example – Variance (2/2)
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Theorems Related to Variance (1/2)
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Theorems Related to Variance (2/2)
Y  a 2  X  2ab X  b 2
2
2
Var[Y ]  E[Y 2 ] - E 2 [Y ]  Y 2  E 2 [aX  b]
C.E.,
(a 2NCU,
 X 2Taiwan
 2ab X  b 2 )  (a 2  X2  2ab X  b 2 )  a 2 (  X 2   X2 )  a 2 Var22[ X ]
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Example –
Theorems Related to Variance (1/2)
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Example –
Theorems Related to Variance (2/2)
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Outline
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Definitions
Families of Discrete Random Variables
Cumulative Distribution Function (CDF)
Averages
Derived Random Variable
Variance and Standard Deviation
Conditional Probability Mass Function
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Get Unconditional PMF from
Conditional PMF
• The conditioning event B contains information about X but not
the precise value of X.
• Use the law of total probability!
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Example – Get Unconditional PMF from
Conditional PMF (1/2)
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Example – Get Unconditional PMF from
Conditional PMF (2/2)
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Conditional PMF
P[ X  x, B]
PX |B ( x) 
P[ B]
xB
• When we learn that an outcome x  B , the probabilities
of all x  B are 0 in our conditional model.
• The probabilities of all x  B are proportionally higher than
they were before we learned x  B .
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Example – Conditional PMF (1/2)
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Example – Conditional PMF (2/2)
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Conditional Expected Value (1/2)
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Conditional Expected Value (2/2)
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