midterm1 - ORIE 360/560 Engineering Probability and...

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Unformatted text preview: ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 Midterm 1 All problems have equal weight. SHOW YOUR WORK. GOOD LUCK! Problem 1 A biased coin, twice as likely to come up heads as tails, is tossed once. If it shows heads, two chips are drawn from urn I, which contains 3 white chips and 4 red chips; if it shows tails, two chips are drawn from urn II, which contains 6 white chips and 3 red chips. Given that both a white chip and a red chip were drawn, what is the probability that the coin came up tails? Problem 2 The pdf of a continuous random variable X is given by 8 if 0 < x < 1 >2 <a if 2 < x < 3 f (x) = 1 (x 3) if 3 < x < 4 : >2 :0 otherwise (a) Find a and plot the density. (b) Compute and plot the cumulative distribution function of X . (c) Find P (X 2), P (X = 1) and P (jX 2j > 1). Problem 3 A continuous random vector (X; Y ) has a joint pdf given by ( ) e if 0 < y < 1 and x > y : f (x; y) = 0 otherwise (a) Find the marginal probability densities of X and Y (caution: two cases for the marginal density of X ) (b) Are X and Y independent? (c) Compute the probability P (X + Y 2). Problem 4 The amount X in pounds of polyurethane cushioning found in a car is modeled as a continuous random variable with pdf 1 1 if 25 x 50 : ln 2 f (x) = 0 otherwise (a) Find the mean, variance and standard deviation for X . (b) Let Y = 1=X . Find the mean of Y . a X x y X;Y X x 1 ...
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