Expectation Theorem CDF Revisited Multiple Continuous Random Variables

Expectation theorem cdf revisited multiple continuous

This preview shows page 10 - 20 out of 35 pages.

Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 6/25 Conditional PDF on an Event
Image of page 10
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 6/25 Conditional PDF on an Event
Image of page 11
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 7/25 Conditional Expectations on an Event Discrete Random Variables 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 )
Image of page 12
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 7/25 Conditional Expectations on an Event Discrete Random Variables 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 ) Continuous Random Variables The conditional expectation of X given an event A with P ( A ) > 0 , is defined by E [ X | A ] = Z xf X | A ( x ) dx
Image of page 13
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 8/25 Expected Value Rule Discrete Random Variables For a function g ( X ) E [ g ( X ) | A ] = X x g ( x ) p X | A ( x )
Image of page 14
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 8/25 Expected Value Rule Discrete Random Variables For a function g ( X ) E [ g ( X ) | A ] = X x g ( x ) p X | A ( x ) Continuous Random Variables For a function g ( X ) E [ g ( X ) | A ] = Z g ( x ) f X | A ( x ) dx
Image of page 15
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 9/25 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 ) = ? P ( B ) = n X i =1 P ( A i ) P ( B | A i ) p X ( x ) = n X i =1 P ( A i ) p X | A i ( x )
Image of page 16
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 9/25 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 ) = ? P ( B ) = n X i =1 P ( A i ) P ( B | A i ) p X ( x ) = n X i =1 P ( A i ) p X | A i ( x ) Definition f X ( x ) = n X i =1 P ( A i ) f X | A i ( x )
Image of page 17
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited Multiple Continuous Random Variables Conditional PDF on a Random Variable 10/25 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 ]
Image of page 18
Probability & Statistics Conditional PDF on an Event Conditional PDF Total Expectation Theorem CDF: Revisited
Image of page 19
Image of page 20

You've reached the end of your free preview.

Want to read all 35 pages?

  • Fall '14
  • LuisDavidGarcia-Puente
  • Probability theory, CDF, random variable X, value Y

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes