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session2 - EE278 Review Session 2 Han-I Su Han-I Su EE278...

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EE278: Review Session # 2 Han-I Su April 13, 2009 Han-I Su () EE278: Review Session # 2 April 13, 2009 1 / 12

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Outline 1 Administrative Information 2 CDF, PMF, and PDF 3 Functions of a Random Variable 4 Homework Hints Han-I Su () EE278: Review Session # 2 April 13, 2009 2 / 12
Administrative Information Instructor office hours Time: MW 1:30-2:30 Location: Packard 269 TA office hours Time: W 3-5 (regular) Location: Packard 106 Phone: 725-1739 Han-I Su () EE278: Review Session # 2 April 13, 2009 3 / 12

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Properties of CDF F X ( x ) = P { X x } ≥ 0 F X ( a ) F X ( b ) if a > b lim x →−∞ F X ( x ) = 0 , lim x + F X ( x ) = 1 F X ( a + ) = F X ( a ) P { X = a } = F X ( a ) F X ( a ) Han-I Su () EE278: Review Session # 2 April 13, 2009 4 / 12
Properties of CDF F X ( x ) = P { X x } ≥ 0 F X ( a ) F X ( b ) if a > b lim x →−∞ F X ( x ) = 0 , lim x + F X ( x ) = 1 F X ( a + ) = F X ( a ) P { X = a } = F X ( a ) F X ( a ) F X ( x ) has at most countably many jumps, that is, { a : F X ( a ) > F X ( a ) } is countable Proof : Han-I Su () EE278: Review Session # 2 April 13, 2009 4 / 12

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Properties of CDF F X ( x ) = P { X x } ≥ 0 F X ( a ) F X ( b ) if a > b lim x →−∞ F X ( x ) = 0 , lim x + F X ( x ) = 1 F X ( a + ) = F X ( a ) P { X = a } = F X ( a ) F X ( a ) F X ( x ) has at most countably many jumps, that is, { a : F X ( a ) > F X ( a ) } is countable Proof : What is the maximum number of jumps 1 / 2? Han-I Su () EE278: Review Session # 2 April 13, 2009 4 / 12
Properties of CDF F X ( x ) = P { X x } ≥ 0 F X ( a ) F X ( b ) if a > b lim x →−∞ F X ( x ) = 0 , lim x + F X ( x ) = 1 F X ( a + ) = F X ( a ) P { X = a } = F X ( a ) F X ( a ) F X ( x ) has at most countably many jumps, that is, { a : F X ( a ) > F X ( a ) } is countable Proof : What is the maximum number of jumps 1 / 2? Ans: |{ a : F X ( a ) F X ( a ) 1 / 2 }| ≤ F X (+ ) F X ( −∞ ) 1 / 2 = 2 Han-I Su () EE278: Review Session # 2 April 13, 2009 4 / 12

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Properties of CDF F X ( x ) = P { X x } ≥ 0 F X ( a ) F X ( b ) if a > b lim x →−∞ F X ( x ) = 0 , lim x + F X ( x ) = 1 F X ( a + ) = F X ( a ) P { X = a } = F X ( a ) F X ( a ) F X ( x ) has at most countably many jumps, that is, { a : F X ( a ) > F X ( a ) } is countable Proof : What is the maximum number of jumps 1 / 2? Ans: |{ a : F X ( a ) F X ( a ) 1 / 2 }| ≤ F X (+ ) F X ( −∞ ) 1 / 2 = 2 All jumps: uniontext k =2 { jumps 1 / k } (countable union of finite sets) Han-I Su () EE278: Review Session # 2 April 13, 2009 4 / 12
Example P { X = 2 } P { X < 2 } P( { X = 2 } ∪ { 0 . 5 X 1 . 5 } ) P( { X = 2 } ∪ { 0 . 5 X 3 } ) F X ( x ) 1 1 2 3 4 1 / 3 2 / 3 x 1 3 x 2 Han-I Su () EE278: Review Session # 2 April 13, 2009 5 / 12

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Example P { X = 2 } P { X < 2 } P( { X = 2 } ∪ { 0 . 5
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session2 - EE278 Review Session 2 Han-I Su Han-I Su EE278...

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