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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Part III Joint Distribution 11/10/2009 1 Discrete Case 1. Joint distribution (joint pmf) De nition and notation: f ( x,y ) = P ( X = x,Y = y ) Properties: (1) f ( x,y ) ≥ , (2) ∑ x ∑ y f ( x,y ) = 1 A distribution table/matrix 2. Marginal distribution: The distributions of X and Y , when considered separately. De nition f X ( x ) = P ( X = x ) = ∑ y f ( x,y ) f Y ( y ) = P ( Y = y ) = ∑ x f ( x,y ) Connection with distribution table: The marginal distributions f X ( x ) and f Y ( y ) can be obtained from the distribution table as the row sums and column sums of the entries. These sums can be entered in the margins of the table as an additional column and row. 3. Expectation and variance μ X , μ Y , σ 2 X , σ 2 Y denote the (ordinary) expectations and variances of X and Y . They can be computed as usual: μ X = ∑ x xf X ( x ) , or computed from the joint pmf directly: μ X = ∑ x ∑ y xf ( x,y ) . Expectation of a function of X and Y : E ( μ ( X,Y )) = ∑ x ∑ y μ ( x,y ) f ( x,y ) . 4. Probability calculation Probabilities involving X and Y (e.g., P ( X + Y = 3) or P ( X ≥ Y ) can be computed by adding up the corresponding entries in the distribution table. 5. Covariance and correlation De nition Covariance of X and Y : Cov ( X,Y ) = E ( X- μ X )( Y- μ Y ) = E ( XY )- E ( X ) E ( Y ) Correlation of X and Y : ρ ( X,Y ) = Cov ( X,Y ) σ X σ Y Properties: covariance can be positive or negative,- 1 ≤ ρ ( X,Y ) ≤ 1 . Relation to variance: V ar ( X ) = Cov ( X,X ) V ar ( X + Y ) = V ar ( X ) + V ar ( Y ) + 2 Cov ( X,Y ) V ar ( X- Y ) = V ar ( X ) + V ar ( Y )- 2 Cov ( X,Y ) Cov ( aX,bY ) = ab · Cov ( X,Y ) 1 Cov ( W + X,Y + Z ) = Cov ( W,Y ) + Cov ( W,Z ) + Cov ( X,Y ) + Cov ( X,Z ) 6. Independence of random variables De nition: X and Y are called independent if the joint pmf is the product of the indi- vidual pmfs, i.e., if f ( x,y ) =...
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This note was uploaded on 08/08/2010 for the course AMS 410 taught by Professor Yang,y during the Fall '08 term at SUNY Stony Brook.
- Fall '08