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09_26 - Y(Exam 2 follow a bivariate normal distribution...

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STAT 410 Examples for 09/26/2011 Fall 2011 Bivariate Normal Distribution: ( ) ( ) - - - - = 2 1 1 2 2 2 1 σ μ ρ ρ σ σ π 1 2 1 1 2 1 , exp x y x f - + - - - 2 2 2 2 2 1 1 σ μ σ μ σ μ ρ 2 y y x , - < x < , - < y < . (a) the marginal distributions of X and Y are 2 1 , 1 σ μ N and 2 2 , 2 σ μ N , respectively; (b) the correlation coefficient of X and Y is ρ XY = ρ , and X and Y are independent if and only if ρ = 0; (c) the conditional distribution of Y, given X = x , is ( ) ( ) - - + 2 2 2 , 1 1 2 2 σ ρ μ σ σ ρ μ 1 x N ; (d) the conditional distribution of X, given Y = y , is ( ) ( ) - - + 2 1 2 , 2 2 1 1 σ ρ μ σ σ ρ μ 1 y N . (e) a X + b Y is normally distributed with mean E ( a X + b Y ) = a μ 1 + b μ 2 and variance Var ( a X + b Y ) = a 2 σ 1 2 + 2 a b ρ σ 1 σ 2 + b 2 σ 2 2 .
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ρ = 0.0 ρ = 0.3 ρ = 0.6 ρ = 0.9
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1. A large class took two exams. Suppose the exam scores X (Exam 1) and
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Unformatted text preview: Y (Exam 2) follow a bivariate normal distribution with μ 1 = 70, σ 1 = 10, μ 2 = 60, σ 2 = 15, ρ = 0.6. a) A students is selected at random. What is the probability that his/her score on Exam 2 is over 75? b) Suppose you're told that a student got a 80 on Exam 1. What is the probability that his/her score on Exam 2 is over 75? c) Suppose you're told that a student got a 66 on Exam 1. What is the probability that his/her score on Exam 2 is over 75? d) Suppose you're told that a student got a 70 on Exam 2. What is the probability that his/her score on Exam 1 is over 80? e) A students is selected at random. What is the probability that the sum of his/her Exam 1 and Exam 2 scores is over 150? f) What proportion of students did better on Exam 1 than on Exam 2?...
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