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Assignment5

# Assignment5 - 1 Assignment 5 STAT220-Winter 2011 Instructor...

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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 ) . (b)- Let α be a real number. Determine the value of α such that X 1 and X 2 + αX 3 are uncorrelated. (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 . Problem 2- Let X be a random variable whose moment generating function is M X ( t ) = a 2 - e - t . where a . (a)- Find a . (b)- Determine values of t such that the definition for M X ( t ) is meaningful. (c)- Find E ( X ( X 2 + 1)) . (d)- Find f X ( . ) . (e)- Find P ( X is an even number ) . Problem 3 - In this problem we are going to see why the sum of two independent random variables X 1 ∼ N ( μ 1 , σ 2 1 ) and X 2 ∼ N ( μ 2 , σ 2 2 ) is again a normal random variable. (a)- Recall that for two independent discrete random variables W 1 and W 2 we showed in class that E [ g ( W 1 ) h ( W 2 )] = E [ g ( W 1 )] E [ h ( W 2 )] holds for “any” two functions g ( . ) and h ( . ) . In fact, the same statement holds for any two independent continuous random variables Z 1 and Z 2 . Use this fact to show that M Z 1 + Z 2 ( t ) = M Z 1 ( t ) M Z 2 ( t ) .

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2 (b)- Use the statement proved in part (a) to show that X 1 + X 2
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Assignment5 - 1 Assignment 5 STAT220-Winter 2011 Instructor...

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