sampleexamjointdistributions

# sampleexamjointdistributions - (b Provide the conditional...

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STA 4321/5325 Spring 2010 Sample Exam: Joint Distributions Full Name: On my honor, I have neither given nor received unauthorized aid on this examination. Signature: This is a 50 minute exam. There are 4 problems, worth a total of 40 points. Remember to show your work. Answers lacking adequate justi±cation may not receive full credit. You may use one letter-sized sheet (the same size as the lecture notes) of your own notes and a pocket calculator. (You are not required to bring a calculator | you may leave your answers in a form from which the numerical answer could be immediately calculated.) You may not use any books, other references, or text-capable electronic devices. 1. The proportions X and Y of two chemicals found in samples in an insecticide have the joint probability density function f X;Y ( x;y ) = ( 2 if 0 ± x ± 1 ; 0 ± y ± 1 ; 0 ± x + y ± 1 ; 0 otherwise : (a) Provide the marginal probability density function of Y at 0 : 7, i.e., provide f Y (0 : 7). [2 pts]

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Unformatted text preview: (b) Provide the conditional probability density function of X given Y = 0 : 7. [4 pts] (c) Provide P ( X > : 95 j Y = 0 : 7). [4 pts] 2. Suppose we have two random variables X and Y with joint probability density function f X;Y ( x;y ) = ( e ± x if 0 ± y ± x < 1 ; otherwise : (a) Provide P ( X < 2 ;Y > 1). [2 pts] (b) Provide P ( X ² 2 Y ). [4 pts] (c) Provide the marginal density fucntion of X . [4 pts] 3. Consider X and Y with joint density function as in Problem 1. (a) Provide E [ X ]. [2 pts] (b) Provide E [ X + Y ]. [4 pts] (c) Proivde V ( X + Y ). [4 pts] Hint: Note that E [ g ( X;Y )] = R 1 ±1 R 1 ±1 g ( x;y ) f X;Y ( x;y ) dxdy . 4. Let X and Y be independent random variables with E [ X ] = 56 ;V ( X ) = 16, and E [ Y ] = 5 ;V [ Y ] = 4. (a) Provide Cov ( X;Y ). [2 pts] (b) Provide V ( X + Y ). [4 pts] (c) Provide V ( X ± Y ). [4 pts]...
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sampleexamjointdistributions - (b Provide the conditional...

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