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samplemidterm2

# samplemidterm2 - AMS 310 Survey of Probability and...

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AMS 310: Survey of Probability and Statistics MIDTERM II Spring 2009 Last Name: First Name: ID: Show all your work for full credit 1. Let X be a continuous random variable having the probability density function f ( x ) = 1 - | x | if - c x c , 0 otherwise. where c > 0 is a constant. (a) (2pts) Plot the function f ( x ). (b) (3pts) Find c . (b) (3pts) Find the variance V ar ( X ). (d) (4pts) Determine the first quartile Q 1 and the median Q 2 of X . 2. Let X and Y be two continuous r.v.’s with joint density function f ( x, y ) = 4 9 (2 + xy ) if 0 x 1, 0 y 1, 0 otherwise 1

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(a) (5pts) Find the marginal distribution f X ( x ). (b) (5pts) Find P ( X 1 2 ). (c) (5pts) Find the probability P ( X 1 2 , Y 1 2 ). (d) (5pts) Find the conditional probability P ( X > 1 2 | Y > 1 2 ). 3. Let W be a continuous random variable having exponential distribution with parameter λ = 2 . 5. (a) (4pts) Find the mean μ and variance σ 2 of W . (b) (4pts) Find P ( W > 6 . 5 | W > 2 . 5). 4. Suppose that X and Y are two discrete r.v.’s with joint mass function f ( x, y ) = a ( x + y ) for x = 0 , 1 , 2 and y = 1 , 2 , 3 0 otherwise 2
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samplemidterm2 - AMS 310 Survey of Probability and...

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