week03-1b

week03-1b - TA session 3 Econ. 103, winter 2010 Wed., Jan....

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Econ. 103, winter 2010 Wed., Jan. 20, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E. 1 Equations shortlist for HW1 1.1 One normal random variable Let X ∼ N ( μ x , σ 2 x ) As you know: E [ X ] = μ x , Var[ X ] = σ 2 x , Stddev[ X ] = σ x The mean shifts under addition: X + a ∼ N ( μ x + a, σ 2 x ) The variance gains a coefficient under multiplication: bX ∼ N ( μ x , b 2 σ 2 x ) So bX + a ∼ N ( μ x + a, b 2 σ 2 x ) . Useful : to make a normal random variable standard normal: X - μ x σ x ∼ N (0 , 1) 1.2 Two normal random variables Let X ∼ N ( μ x , σ 2 x ), and Y ∼ N ( μ y , σ 2 y ) The sum of any number of normal random variables is also normally distributed. E [ X + Y ] = E [ X ] + E [ Y ] Var[ X + Y ] = Var[ X ]+ Var[ Y ]+ 2Cov[ X,Y ] Cov[ X,Y ] = E [( X - μ x )( Y - μ y )], which can be rewritten Cov[ X,Y ] = E [ X · Y ] - μ x μ y Meaning: X + Y ∼ N ( μ x + μ y , σ 2 x + σ 2 y + 2Cov[ X,Y ] . Additionally,
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week03-1b - TA session 3 Econ. 103, winter 2010 Wed., Jan....

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