Test 3 Solutions (Fall 2010)

Test 3 Solutions (Fall 2010) - 1 NAME ISyE 3770 Solutions...

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1 NAME ISyE 3770 Solutions — Test 3 — Fall 2010 This test is 55 minutes long. You are allowed three cheat sheets. 1. If U 1 , U 2 , U 3 are i.i.d. Unif(0,1), what’s the distribution of - (1 / 3) 3 i =1 n( U i )? Solution: Erlang 3 (3) 2. If X 1 , X 2 , X 3 are i.i.d. Pois(0 . 2), what’s the distribution of 3 i =1 X i ? Solution: Pois(0.6) 3. If X is standard normal, what’s the probability that X < - 1? Solution: P ( X < - 1) = Φ( - 1) = 1 - Φ(1) = 0 . 1587 . 4. Suppose that the rainfall during November is Nor(10 , 4) (measured in centimeters). Find the probability that November rainfall will total at least 14 cm. Solution: P ( X > 14) = P ( X - 10 4 > 14 - 10 4 ) = P ( X > 2) = 1 - Φ(2) = 0 . 0227 . 5. If X and Y are i.i.d. Nor(0 , 1), what’s the probability that X - Y > 1? Solution: Because X Nor(0 , 1) and Y Nor(0 , 1), X - Y follows a Normal distribution with E [ X - Y ] = 0 and Var ( X - Y ) = Var ( X ) + Var ( Y ) = 2, i.e., X - Y Nor(0 , 2). Thus, P ( X - Y > 1) = P ( X - Y 2 > 1 - 0 2 ) = 1 - Φ(0 . 707) = 0 . 24 . 6. If X and Y are Nor(0 , 1) and X + Y Nor(0 , 1), what’s the correlation of X and Y ? Solution: Var ( X + Y ) = Var ( X ) + Var ( Y ) + 2 · Cov ( X, Y ) = 2 + 2 · Cov ( X, Y ) = 1.
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