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HW2-2 - Question 7 Consider f X x and f Y y as follows f X...

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ECE 302, Homework #2. Due date: 1/26 http://www.ece.purdue.edu/ ∼ chihw/ECE302 07.html Prof. Chih-Chun Wang Email: [email protected] Office: MSEE354 Office Hours: MWF: 12pm–1pm TA: Kamesh Krishnamurthy, Email: [email protected] Office: POTR 370 Office Hours: M: 10–11am TTh: 10:30am–12:30pm Review of Calculus: Question 1: Define a 2-D function f ( x, y ) as follows. f ( x, y ) = ( 1 x e - y x if x ∈ (0 , 1] and y ∈ [0 , ∞ ) 0 otherwise Compute the value of the following 2-dimensional integral. Z ∞ x = -∞ Z ∞ y = -∞ xyf ( x, y ) dydx. Question 2: Consider a best-of-three baseball series between two teams A and B , namely, the team that wins 2 games wins the whole series, and no game will be played after that since the series is determined. 1. What is the sample space? 2. What is the weight assignment you would like to make? Is your weight assignment a reasonable one? Question 3: Problem 2.13. (a,c,e). Question 4: Problem 2.19. Question 5: Let A, B, C ⊆ S . Prove the following DeMorgan’s Rule: ( A ∪ B ∪ C ) c = A c ∩ B c ∩ C c .
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Hint: by an iterative approach or by definition. Question 6: Problem 2.26. (Use the result in Problem 2.25(a).)
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Unformatted text preview: Question 7: Consider f X ( x ) and f Y ( y ) as follows. f X ( x ) = ( e-x if x ≥ otherwise f Y ( y ) = 0 . 5 e-| y | Compute the following two integrals Z ∞ y =-∞ Z ∞ x =-∞ ( x + y ) f X ( x ) f Y ( y ) dxdy Z ∞ y =-∞ Z ∞ x =-∞ e-. 1( x + y ) f X ( x ) f Y ( y ) dxdy Question 8: Problem 2.28. Answer the following questions first. 1. What is the sample space? 2. What is the corresponding weight assignment? Hint: It is a continuous sample space so you have to use a curve f X ( x ) to describe your weight assignment Question 9: (It is actually Problem 2.30.) For any valid weight assignment over a contin-uous sample space, answer the following questions. 1. Show that we must have P ( X ∈ (-∞ ,r ]) ≤ P ( X ∈ (-∞ ,s ]) if r < s . 2. Suppose we know the values of P ( X ∈ (-∞ ,r ]) and P ( X ∈ (-∞ ,s ]). What is the value of P ( X ∈ ( r,s ]). Question 10: Problem 2.51....
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