hw1 - ce- 2 x x otherwise be the pdf of a random variable X...

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IE 336 Handout #2 Jan. 21, 2011 Due Jan. 28, 2011 Homework Set #1 1. Let B 1 ,B 2 ,...,B n be a partition of the sample space of a random experiment. Using the axioms 1 - 3 show that P ( A ) = n X i =1 P ( A | B i ) P ( B i ) where A is an event from the sample space. ( Hint: Try considering A ( B 1 B 2 ∪···∪ B n ) = ( A B 1 ) ( A B 2 ) ∪ ··· ∪ ( A B n ).) 2. There is a box with ten black balls, numbered 1 , 2 , 4 , 8 , 10 , 13 , 15 , 18 , 20 , 21, and eight white balls numbered 3 , 5 , 6 , 11 , 14 , 17 , 19 , 22. One ball from the box is selected at random. Let A , B , and C denote the events in the following way: A: number of the selected ball is less than 8 B: odd-numbered ball is selected C: selected ball is white and even-numbered. (a) Determine A , B , and C . (b) Are events A and B independent? Explain. (c) Are events A and C independent? Explain. 3. Let f ( x ) = ce - 5 x - 3 x
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Unformatted text preview: ce- 2 x x otherwise be the pdf of a random variable X . Determine the value of the constant c . Find E ( X ). 4. Let f ( x,y ) = ( 2 e-x-y y x < otherwise be the joint pdf of two random variables X and Y . Find f ( x | y ) and f ( y | x ). 5. The joint pdf of two random variables X and Y is given by the following table. X \ Y 3 7 15 20 2 a . 1 1 . 5 b . 05 5 . 075 1 . 5 a . 075 . 3 b For example P ( X = 5 ,Y = 15) = 0 . 075. Also let P ( X = 5) = 0 . 5. (a) Determine the values of constants a and b . (b) Find P ( X = 5 | Y = 7) and P ( X = 2 | Y = 15). (c) Find E ( X | Y = 20). (d) Determine E ( Y ). 1...
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