Lecture7.pdf - Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables ECE 2521

Lecture7.pdf - Conditional Distribution for Two Random...

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Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables ECE 2521: Analysis of Stochastic Processes Lecture 7 Department of Electrical and Computer Engineering University of Pittsburgh October, 10 th 2018 Mircea Lupu, PhD Lecture 7 ECE 2521: Analysis of Stochastic Processes
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Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables Conditioning by an Event Conditioning by a Random Variable Law of Iterated Expectations (Total Expectation Theorem) Conditioning by an Event We consider the probability model for two or more random variables given the knowledge that some event B has occured Conditional Joint Probability Mass Function For two discrete random variables X and Y and a conditioning event B (Prob( B ) > 0) the conditional joint PMF of X and Y : p X , Y | B ( x ) = Prob( X = x and Y = y | B ) = ( p X , Y ( x , y ) Prob( B ) if ( x , y ) B 0 otherwise . The conditional joint PMF is non-zero for a pair ( x , y ), if ( x , y ) is contained in the conditioning event B , and zero otherwise Satisfies the axioms of probability: (1) Non-negativity: p X , Y | B ( x , y ) 0. (2) Normalization: ( x , y ) B p X , Y | B ( x , y ) = 1 Lecture 7 ECE 2521: Analysis of Stochastic Processes
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Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables Conditioning by an Event Conditioning by a Random Variable Law of Iterated Expectations (Total Expectation Theorem) Conditioning by an Event Conditional Joint Probability Density Function For two continuous random variable X and Y and a conditioning event B (Prob( B ) > 0) the conditional joint PDF of X and Y : f X , Y | B ( x , y ) = ( f X , Y ( x , y ) Prob( B ) if ( x , y ) B 0 otherwise . Satisfies the axioms of probability: (1) Non-negativity: f X , Y | B ( x , y ) 0. (2) Normalization: R -∞ R -∞ f X , Y | B ( x , y ) dxdy = 1 Lecture 7 ECE 2521: Analysis of Stochastic Processes
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Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables Conditioning by an Event Conditioning by a Random Variable Law of Iterated Expectations (Total Expectation Theorem) Conditional Expectations The conditional expected value of a function of two random variables W = g ( X , Y ) given condition B is: E [ W | B ] = E [ g ( X , Y ) | B ] = ( x , y ) B g ( x , y ) p X , Y | B ( x , y ) if X , Y discrete R -∞ R -∞ g ( x , y ) f X , Y | B ( x , y ) dxdy if X , Y continuous Then conditional variance of W is computed by: Var [ W | B ] = E [ W 2 | B ] - ( E [ W | B ]) 2 . Lecture 7 ECE 2521: Analysis of Stochastic Processes
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Conditional Distribution for Two Random Variables Independence for Two Random Variables Functions of Two Random Variables Conditioning by an Event Conditioning by a Random Variable Law of Iterated Expectations (Total Expectation Theorem) Example 1 (Discrete) Random variables L and T have the following joint PMF: P L , T ( l , t ) t = 40 sec t = 60 sec l = 1 page 0.15 0.1 l = 2 pages 0.3 0.2 l = 3 pages 0.15 0.1 For the random variable V = LT , we define the event: A = { V > 90 } Find the following: (1) the conditional PMF p L , T | A ( l , t ) (2) E [ V | A ] (3) Var[ V | A ].
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  • Fall '09
  • Jacobs
  • Probability theory, conditional distribution, continuous random variable X, X given Y, Analysis of Stochastic Processes

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