test4 - X N ( , 2 ) where beta(2 , 2). Compute E [ X ] and...

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Fall 2008 Test IV Page: 1 of 1 Introduction to Probability Monday, November 17, 2008 STA 4321/5325 Instructions: Please turn of your cell phones. Please write all oF your answers on a separate sheet oF paper and make sure you have clearly labeled the problem corresponding to your answer. Absolutely no cheating. This test has a total oF 100 points. Work as quickly and e±ciently as possible so that you can ²nish all oF the problems. Name: Some Equations Beta If X beta( α,β ), the pdf of X is f ( x )= ± Γ( α + β ) Γ( α )Γ( β ) x α - 1 (1 - x ) β - 1 , 0 <x< 1 0 , else Also, E[ X ]= α α + β and var( X )= αβ ( α + β ) 2 ( α + β +1) . Multinomial If X multinomial( n ; p 1 , . . . , p k ), then X has the pmf P ( X 1 = x 1 , . . . , X k = x k )= n ! x 1 ! · · · x k ! p x 1 1 · · · p x k k Also, E( X i )= np i , var( X i )= np i q i , and cov( X i ,X j )= - np i p j ( i ± = j ). 1 (12 points) Calculate the following integral. You must show your work. (“I put it in my calculator” will receive 0 points.) The calculation should be rather quick if you utilize knowledge from this course. ² 1 0 x 2 (1 - x ) 7 dx 2 (10+10=20 points) Suppose
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Unformatted text preview: X N ( , 2 ) where beta(2 , 2). Compute E [ X ] and var( X ). 3 (8+8+8+8+8+8=48 points) Suppose ( X, Y ) had the following joint pdf f ( x, y ) = cxy, x 1 , y 1 , else (a) Find c . (b) Compute the marginal distributions for X and Y . (c) Are X and Y independent? (d) Compute cor( X, Y ). (e) Compute the conditional distribution of X | Y . (f) Compute E[ X | Y ]. 4 (10+10=20 points) Each of ve balls are placed in one of four boxes. A ball is placed in box A with probability .1, it is placed in box B with probability .2, it is placed in box C with probability .3, and it is placed in box D with probability .4. Compute the following probabilities. (a) Three balls are placed in box A and two balls are placed in box D. (b) Each box has at least one ball in it....
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