15_Exam2_review_Spring_2012

# e x y y 2 2 x p x y x y xp x y

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Unformatted text preview: Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x 2 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ = ∑ ( x − E [ X | Y = y ]) p X |Y ( x | y) ⎣ ⎦x Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x 2 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ = ∑ ( x − E [ X | Y = y ]) p X |Y ( x | y) ⎣ ⎦x = E ⎡ X 2 | Y = y ⎤ − ( E [ X | Y = y ]) ⎣ ⎦ 2 Ilya Pollak CondiAonal means and variances E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x x x p X ,Y ( x, y ) 1 = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x 2 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ = ∑ ( x − E [ X | Y = y ]) p X |Y ( x | y) ⎣ ⎦x = E ⎡ X 2 | Y = y ⎤ − ( E [ X | Y = y ]) ⎣ ⎦ 2 ⎛ ⎞ = ∑ x 2 p X |Y ( x | y) − ⎜ ∑ xp X |Y ( x | y)⎟ ⎝x ⎠ x 2 Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x 2 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ = ∑ ( x − E [ X | Y = y ]) p X |Y ( x | y) ⎣ ⎦x = E ⎡ X 2 | Y = y ⎤ − ( E [ X | Y = y ]) ⎣ ⎦ 2 ⎛ ⎞ 2 = ∑ x p X |Y ( x | y) − ⎜ ∑ xp X |Y ( x | y)⎟ ⎝x ⎠ x 2 E [ X | A ] = ∑ xp X | A ( x ) x 2 2 var( X | A) = E ⎡( X − E [ X | A ]) | A ⎤ =...
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