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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 ] = -∞ 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) V ar [ 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 ] = x -∞ 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 ) = -∞ f X,Y ( x, y ) dy and f Y ( y ) = -∞ 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 ] = g ( x,y ) w f X,Y ( x, y ) dx dy 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 ] = -∞ -∞ 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 [ Y ] = r X,Y - μ X μ Y (d) (section 4.7) Variance of X + Y : V ar [ X + Y ] = V ar [ X ] + V ar [ Y ] + 2 E [( X - μ X )( Y - μ Y )] = V ar [ X ] + V ar [ Y ] + 2 Cov [ X, Y ] (e) (section 4.7) Correlation Coefficient of X and Y : ρ X,Y = Cov [ X, Y ] V ar [ X ] V ar [ Y ] = Cov [ X, Y ] σ X σ Y (f) (section 4.7) Uncorrelated:
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