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# handout_week6_a - Example P X 2 Y 2< 20 = Pairs of...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Conditional Expected Values Given that the expected value of a function of a rv is given by E [ Y ] = Z -∞ yf Y ( y ) dy , we obtain the conditional expectation of Y given X = x as E [ Y | x ] = Z -∞ yf Y ( y | x ) dy . More generally, the conditional mean of g ( Y ) is deﬁned as E [ g ( Y ) |A ] = Z -∞ g ( y ) f Y ( y |A ) dy .

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Example The joint pdf of ( X , Y ) is given by f X , Y ( x , y ) = 6 ( 1 - x - y ) for values of x and y for which ( x , y ) lies within the triangle as shown in the ﬁgure below. Find the conditional expected values of X and X 2 given Y = y . Pairs of Discrete Random Variables The joint probability mass function of ( X , Y ) specify the probabilities of the event { X = x k } ∩ { Y = y k } : p X , Y ( x j , y k ) = P ±² X = x j ³ ∩ { Y = y k } ´ = P ± X = x j , Y = y k ´ j = 1 , 2 , . . . k = 1 , 2 , . . . Thus the joint pmf gives the probability of the occurrence of the pairs ( x j , y k ) . The probability of any event A is the sum of the pmf over the outcomes in A : P [( X , Y ) A ] = XX ( x j , y k ) A p X , Y ( x j , y k ) .
Pairs of Discrete Random Variables

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Unformatted text preview: Example: P [ X 2 + Y 2 < 20 ] = Pairs of Discrete Random Variables • The fact that the probability of the sample space S is 1 gives ∞ X j = 1 ∞ X k = 1 p X , Y ( x j , y k ) = 1 . • The marginal probability mass function of X is obtained by: p X ( x j ) = P [ X = x j ] = P [ X = x j , Y = anything ] = ∞ X k = 1 p X , Y ( x j , y k ) Similarly, p Y ( y k ) = ∞ X j = 1 p X , Y ( x j , y k ) • Conditional expected values: E [ Y | x ] = X y j y j p Y ( y j | x ) . Pairs of Discrete Random Variables For all nonnegative integers k , m , let P [ X = k , Y = m ] = ( 1-v 1 ) ( 1-v 2 ) v k 1 v m 2 ; X and Y might be the numbers of photoelectrons counted at two photodetectors. Find the marginal probability mass function of X ....
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handout_week6_a - Example P X 2 Y 2< 20 = Pairs of...

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