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StatORJune08

# StatORJune08 - Summer 2008 Probability Solve any three...

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Summer 2008 Probability Solve any three problems for a full credit. Indicate below EXACTLY which three problems you have attempted by circling the appropriate numbers: 1 2 3 4 NAME:________________________________________________ SOLAR ID#:___________________________________________ Start your answer on each question sheet. Attach all extra sheets you use to the appropriate sheet. Hand in all question sheets. Time of Exam is from 9AM-12noon

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1. A stick of length 1 is cut into three pieces of lengths X, Y , and Z , with X Y Z . First, the stick of length 1 is cut into two pieces at a randomly chosen point uniformly distributed along its length. Then the smaller of these two sticks is also cut into two pieces at a randomly chosen point uniformly distributed along its length. These two cuts are conducted independently. Find Cov( X,Y ).
2. Let be exponentially distributed with mean 1 X 1 µ ; let be exponentially 2 X distributed with mean 2 µ ; and let and be independent. Compute for 1 X 2 X ) | Pr( 2 1 2 1 bX X aX X < < < < b a 0 . What happens in the limit as ? b

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3. Let X and Y be random variables with joint density function f ( x,y ) defined by f ( x,y ) = x + y, when 0 < x < 1 and 0 < y < 1. Find E [ X E [ Y | X ]].
4. Suppose that X and Y are independent and identically distributed positive random variables. Prove that . 1 ] [ Y X E

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Tables of the Normal Distribution Probability Content from -oo to Z Z | 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ----+---------------------------------------------------------------------- 0.0 | 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 | 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 | 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 | 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 | 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 | 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 | 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
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• Fall '08
• Feinberg,E
• Probability theory, probability density function, joint density function, randomly chosen point, Distribution Probability Content

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