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Sample_Final - EL630 Sample Final Exam(The real test will...

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EL630 Sample Final Exam (The real test will contain less questions) 1. Box 1 contains 3 white, 2 red and 5 black balls. Box 2 contains 4 white, 4 red and 2 black balls. At random, we pick up one ball from each box. Find the probability that these two balls are in different color. 2. X is uniformly distributed on [ /2, /2], tan Y X π π - = . Find ( ) f y and draw it. 3. X and Y are independent with exponential densities ( ) ( ), ( ) ( ), 3 . y x X Y f x e u x f y e u y β α α β β α - - = = Find the density and distribution of Z = X+3Y . 4. X and Y are independent with exponential densities ( ) ( ), ( ) ( ). y x X Y f x e u x f y e u y - - = = Find the joint density Z = X+Y and W=X/Y . 5. 2 2 ( , ) 1/ , 1; zero otherwise. XY f x y x y π = + Discuss whether X,Y are uncorrelated and whether X,Y are independent. 6. Find A such that 2 AX is the MS estimate of Y . 7. Prove or disprove that if 1 2 1 2 then ( ) ( ) ( ) 1 A A A P A P A P A + - . 8. Let 1 2 , , , n X X X ××× be independent, identically distributed exponential random variables with common p.d.f.
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