chap2.2 - Joint Distribution • We may be interested in...

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Unformatted text preview: Joint Distribution • We may be interested in probability statements of sev- eral RVs. • Example: Two people A and B both flip coin twice. X : number of heads obtained by A. Y : number of heads obtained by B. Find P ( X > Y ) . • Discrete case: Joint probability mass function: p ( x,y ) = P ( X = x,Y = y ) . – Two coins, one fair, the other two-headed. A ran- domly chooses one and B takes the other. X = braceleftbigg 1 A gets head A gets tail Y = braceleftbigg 1 B gets head B gets tail Find P ( X ≥ Y ) . 1 • Marginal probability mass function of X can be ob- tained from the joint probability mass function, p ( x,y ) : p X ( x ) = summationdisplay y : p ( x,y ) > p ( x,y ) . Similarly: p Y ( y ) = summationdisplay x : p ( x,y ) > p ( x,y ) . 2 • Continuous case: Joint probability density function f ( x, y ) : P { ( X, Y ) ∈ R } = integraldisplay integraldisplay R f ( x, y ) dxdy • Marginal pdf: f X ( x ) = integraldisplay ∞-∞ f ( x, y ) dy f Y ( y ) = integraldisplay ∞-∞ f ( x, y ) dx • Joint cumulative probability distribution function of X and Y F ( a,b ) = P { X ≤ a,Y ≤ b }- ∞ < a,b < ∞ • Marginal cdf: F X ( a ) = F ( a, ∞ ) F Y ( b ) = F ( ∞ , b ) • Expectation E [ g ( X, Y )] : = ∑ y ∑ x g ( x,y ) p ( x, y ) in the discrete case = integraltext ∞...
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chap2.2 - Joint Distribution • We may be interested in...

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