Test 2 Solutions (Fall 2009)

# Test 2 Solutions (Fall 2009) - 1 NAME ISyE 3770 Test 2a...

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1 NAME ISyE 3770 — Test 2a Solutions — Fall 2009 This test is 55 minutes long. You are allowed one cheat sheet. Put your nice, simple answers here... 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15(a). 15(b). 15(c). 15(d). 15(e). 15(f). 15(g). 16. 17. 18. 19. 20.

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2 1. Suppose X has p.d.f. f ( x ) = 2 x , 0 x 1. Find E [ - 2 X - 5]. Solution: E [ X ] = Z 1 0 2 x 2 dx = 2 3 E [ - 2 X - 5] = - 2 E [ X ] - 5 = - 19 3 2. TRUE or FALSE? E [ X 2 ] ( E [ X ]) 2 . Solution: FALSE. ( Var ( X ) 0) 3. Suppose X has m.g.f. M X ( t ) = 0 . 2 e t + 0 . 8. Find Var ( X ). Solution: If M X ( t ) = pe t + q X Bern( p ) Var ( X ) = pq Thus, X Bern(0 . 2) in this example Var ( X ) = 0 . 2 * 0 . 8 = 0 . 16 4. TRUE or FALSE? If the correlation between X and Y is 0, then X and Y are independent. Solution: FALSE. 5. Suppose that X and Y are independent Exponential( λ ) random variables. Find Var ( XY ) (yup — the variance of the product). Solution: Var ( XY ) = E [( XY ) 2 ] - ( E [ XY ]) 2 = E [ X 2 ] E [ Y 2 ] - ( E [ X ]) 2 ( E [ Y ]) 2 (by independence) = ( Var ( X ) + ( E [ X ]) 2 ) 2 - ( E [ X ]) 4 = 1 λ 2 + 1 λ 2 · 2 - 1 λ · 4 = 2 λ 2 · 2 - 1 λ · 4 = 3 λ 4
3 6. What does “i.i.d.” mean?

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