FormulaSheet

FormulaSheet - APMA 3100 Formula Sheet Page 1 1....

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APMA 3100 Formula Sheet Page 1 1. Conditioning a Random Variable Given an Event B with P [ B ] > 0 (a) (section 2.9) Discrete: P X | B ( x )= P X ( x ) P [ B ] x B 0 otherwise (b) (Section 3.8) Continuous: f X | B ( x )= f X ( x ) P [ B ] x B 0 otherwise 2. Conditional Expected Value of a Function of a Random Variable Given an Event B (a) (section 2.9) Discrete: E [ g ( X ) | B ]= ± x B g ( x ) P X | B ( x ) (b) (Section 3.8) Continuous: For x B, E [ g ( X ) | B ]= ² -∞ g ( x ) f X | B ( x ) dx 3. Conditional Variance of a Random Variable Given an Event B (a) (sections 2.9, 3.8 and 4.8) Var [ X | B ]= E [ X 2 | B ] - ( E [ X | B ]) 2 4. Two Variable Joint CDF, PMF and PDF (a) (section 4.1) F X,Y ( x,y )= P [ X x,Y y ]= ² x -∞ ² y -∞ f X,Y ( u,v ) dv du (b) (section 4.2) P X,Y ( x,y )= P [ X = x,Y = y ] (c) (section 4.4) f X,Y ( x,y )= 2 F X,Y ( x,y ) ∂x∂y 5. Marginal PMFs and PDFs (a) (section 4.3) Discrete: P X ( x )= ± y S Y P X,Y ( x,y ) and P Y ( y )= ± x S X P X,Y ( x,y ) (b) (section 4.5) Continuous: f X ( x )= ² -∞ f X,Y ( x,y ) dy and f Y ( y )= ² -∞ f X,Y ( x,y ) dx
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APMA 3100 Formula Sheet Page 2 6. Functions of Two Random Variables W = g ( X,Y ) (a) (section 4.6) PMF of W: P W ( w )= ± ( x,y ): g ( x,y )= w P X,Y ( x,y ) (b) (section 4.6) CDF of W: F W ( w )= P [ W w ]= ²² g ( x,y ) w f X,Y ( x,y ) dxdy 7. Expected Value of W = g ( X,Y ) (a) (section 4.7) Discrete: E [ W ]= ± x S X ± y S Y g ( x,y ) P X,Y ( x,y ) (b) (Section 4.7) Continuous: E [ W ]= ² -∞ ² -∞ g ( x,y ) f X,Y ( x,y ) dx dy 8. Expected Value, Correlation, Covariance, Variance, Correlation Coefficient and Uncorrelated (a) (section 4.7) Expected Value of X + Y : E [ X + Y ]= E [ X ]+ E [ Y ] (b) (section 4.7) Correlation of X and Y : r X,Y = E [ XY ] (c) (section 4.7) Covariance of X and Y : Cov [ X,Y ]= E [( X - μ X )( Y - μ Y )] = E [ XY ] - E [ X ] E
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This note was uploaded on 07/31/2010 for the course APMA 308 taught by Professor Pindera,m during the Spring '08 term at UVA.

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FormulaSheet - APMA 3100 Formula Sheet Page 1 1....

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