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Solutions_to_Assignment5

Solutions_to_Assignment5 - 1 Assignment 5 STAT220-Winter...

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Unformatted text preview: 1 Assignment 5, STAT220-Winter 2011 Instructor: Kamyar Moshksar Due on April 4th, In class Problem 1- Let ( X 1 ,X 2 ,X 3 ) ∼ Mult( n,p 1 ,p 2 ,p 3 ) . (a)- Compute Cov( X 1 + 3 X 2 ,X 2 + X 3 ) . We know that Cov ( X i ,X j ) =- np i p j and V ar ( X i ) =- np i (1- p i ) . Also, Cov ( X 1 + 3 X 2 ,X 2 + X 3 ) = Cov ( X 1 ,X 2 ) + Cov ( X 1 ,X 3 ) + 3 V ar ( X 2 ) + 3 Cov ( X 2 ,X 3 ) . and the solution follows. (b)- Let α be a real number. Determine the value of α such that X 1 and X 2 + αX 3 are uncorrelated. Cov ( X 1 ,X 2 + αX 3 ) = Cov ( X 1 ,X 2 ) + αCov ( X 2 ,X 3 ) =- np 1 p 2- nαp 1 p 3 = 0 → α =- p 2 p 3 . (c)- Calculate ρ X 1 ,X 2 2 . (d)- Write down the formulae for f X 1 ,X 2 | X 1 + X 3 ( .,. | t ) where t ∈ R X 1 + X 3 . f X 1 ,X 2 | X 1 + X 3 ( x 1 ,x 2 | t ) = Pr { X 1 = x 1 ,X 2 = x 2 ,X 1 + X 3 = t } f X 1 + X 3 ( t ) = Pr { X 1 = x 1 ,X 2 = x 2 ,X 3 = t- x 1 } f X 1 + X 3 ( t ) = f X 1 ,X 2 ,X 3 ( x 1 ,x 2 ,t- x 1 ) f X 1 + X 3 ( t ) . But, f X 1 ,X 2 ,X 3 ( x 1 ,x 2 ,t- x 1 ) = n ! x 1 ! x 2 !( t- x 1 )! p x 1 1 p x 2 2 p t- x 1 3 x 2 = n- t, ≤ x 1 ≤ t otherwise and f X 1 + X 3 ( t ) = n !...
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Solutions_to_Assignment5 - 1 Assignment 5 STAT220-Winter...

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