Test 3b Solutions (Fall 2010)

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

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Unformatted text preview: 1 NAME → ISyE 3770 Solutions — Test 3b — 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- 3 ∑ 3 i =1 ‘ n( U i )? Solution: Erlang 3 (1/3) ♦ 2. If X 1 , X 2 , X 3 are i.i.d. Nor(1 , 2), what’s the distribution of ∑ 3 i =1 X i ? Solution: Nor(3,6) ♦ 3. If X is standard normal, what’s the probability that X >- 1? Solution: P ( X >- 1) = Φ(1) = 0 . 8413 . ♦ 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- √ 2 ) = 1- Φ(0 . 707) = 0 . 24 . ♦ 6. If X and Y are Nor(1 , 1) and X + Y ∼ Nor(2 , 1), what’s the correlation of X and Y ?...
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Test 3b Solutions (Fall 2010) - 1 NAME → ISyE 3770...

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