Expectations Total Expectation Theorem Multiple Discrete Random Variables

Expectations total expectation theorem multiple

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Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 7/27 Conditional PMF on an Event
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 8/27 Conditional Expectations on an Event Conditional Expectation The conditional expectation of X given an event A with P ( A ) > 0 , is defined by E [ X | A ] , X x xp X | A ( x )
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 8/27 Conditional Expectations on an Event Conditional Expectation The conditional expectation of X given an event A with P ( A ) > 0 , is defined by E [ X | A ] , X x xp X | A ( x ) Expected Value Rule For a function g ( X ) E [ g ( X ) | A ] = X x g ( x ) p X | A ( x )
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 9/27 Conditional Expectations on an Event Example Let X be the roll of a fair six-sided die A be the event that the roll is an even number Calculate E [ X ] and E [ X | A ] .
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 10/27 Review: Total Probability Theorem Partition into A 1 , A 2 , A 3 Have P ( A i ) for every i Have P ( B | A i ) for every i P ( B ) = ? Definition P ( B ) = n X i =1 P ( A i ) P ( B | A i )
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 10/27 Review: Total Probability Theorem Partition into A 1 , A 2 , A 3 Have P ( A i ) for every i Have P ( B | A i ) for every i P ( B ) = ? Definition P ( B ) = n X i =1 P ( A i ) P ( B | A i ) Definition Consider event B : X = x p X ( x ) = n X i =1 P ( A i ) p X | A i ( x )
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 11/27 Conditioned on an Event Total Expectation Theorem If A 1 , . . . , A n be disjoint events that form a partition of the sample space, with P ( A i ) > 0 for all i , then E [ X ] = n X i =1 P ( A i ) E [ X | A i ]
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 12/27 Mean and Variance of the Geometric The Geometric Geometric with parameter 0 < p 1 p X ( k ) = (1 - p ) k - 1 p
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Probability & Statistics Conditional PMF on an Event Conditional PMF Conditional Expectations Total Expectation Theorem Multiple Discrete Random Variables Conditional PMF on a Random Variable 12/27 Mean and Variance of the Geometric The Geometric Geometric with parameter 0 < p 1 p X ( k ) = (1 - p ) k - 1 p
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Probability & Statistics Conditional PMF on an Event
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  • Fall '14
  • LuisDavidGarcia-Puente
  • Probability theory, value Y

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