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# crib_sheet2 - Crib sheet for Exam II Chapter 5 Joint...

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Crib sheet for Exam II Chapter 5: Joint Probability Distributions The marginal probability mass functions of two discrete rv’s X and Y , denoted by p X ( x ) = P ( X = x ) and p Y ( y ) = P ( Y = y ), respectively are given by: p X ( x ) = y p ( x, y ) p Y ( y ) = x p ( x, y ) The marginal probability density function of two continuous rv’s X and Y , denoted by f X ( x ) and f Y ( y ), respectively are given by: f X ( x ) = -∞ f ( x, y ) dy f Y ( y ) = -∞ f ( x, y ) dx Two r.v. X, Y are independent if for every pair of ( x, y ): Discrete: p ( x, y ) = p X ( x ) · p Y ( y ) Continuous: f ( x, y ) = f X ( x ) · f Y ( y ) Conditional pdf/pmf: f Y | X ( y | x ) = f ( x, y ) f X ( x ) , -∞ < y < p Y | X ( y | x ) = p ( x, y ) p X ( x ) , -∞ < y < Covariance and Correlation coefficient of two rv’s X, Y E [ h ( X, Y )] = x y h ( x, y ) · p ( x, y ) E [ h ( X, Y )] = -∞ -∞ h ( x, y ) · f ( x, y ) dxdy Cov ( X, Y ) = E [( X - μ x )( Y - μ y )] = E [ XY ] - μ x μ y ρ x,y = Corr ( X, Y ) = Cov ( X, Y ) σ X σ Y - 1 ρ x,y 1 Chapter 6: Concepts of Point Estimation A point estimator of ˆ θ is an unbiased estimator of θ if E [ ˆ θ ] = θ for every possible value of θ .

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crib_sheet2 - Crib sheet for Exam II Chapter 5 Joint...

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