Stat225-Hints07 - = F X b-F X a(c The PDF f X x is the derivative of the CDF(d Use the PDF E X = R ∞-∞ x 3 f X x dx 6(a c = 1 16(b For x ≥ 0

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Stat 225 – Problem Set 07 Hints Spring 2009 1. The two properties you need are that a PDF f ( x ) satisfies: f ( x ) 0 for all x and R -∞ f ( x ) dx = 1. 2. (a) c = 1 / 9. (b) A calculus problem: E ( X ) = 2 . 25. (c) P ( | X - 2 . 25 | ≤ 0 . 5) = P (2 . 25 - 0 . 5 X 2 . 25 + 0 . 5). 3. (a) Simple calculus. (b) 1 4 x is a linear function. (c) From the picture, it is clear that X is more likely to be in [1 , 2]. To verify this, use the CDF which, for 1 x 3, is defined as F X ( x ) = x 2 - 1 8 . (d) Use the complementation rule and the CDF. (e) E ( X ) = 2 . 167 and E ( X 2 ) = 5. 4. (a) What’s the simplest type of function there is? Is such a function symmetric . (b) Since f X ( x ) is symmetric about zero, we have R 0 - a f X ( x ) dx = - R a 0 f X ( x ) dx . (c) Use the hint; we know Y is symmetric about zero and also E ( Y ) = E ( X - θ ) = E ( X ) - θ . (d) This random variable is symmetric about a + b 2 . 5. (a) It’s a quadratic. (b) Remember, P ( a X b
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Unformatted text preview: ) = F X ( b )-F X ( a ). (c) The PDF f X ( x ) is the derivative of the CDF. (d) Use the PDF: E ( X ) = R ∞-∞ x 3 f X ( x ) dx . 6. (a) c = 1 . 16. (b) For x ≥ 0, F X ( x ) = 1 . 16(1-e-x ). (c) P ( X > 1) = 1-P ( X ≤ 1). (d) E ( X ) = 0 . 686. 7. (a) For the CDF F X ( x ), need to consider 5 cases for x (b) Set F X ( x ) = 0 . 3 and solve for x . 8. (a) It’s a nice smooth increasing function. (b) CDFs are positive, increasing, approach 0 as x goes to-∞ , and approach 1 as x approaches ∞ . (c) Use the chain rule: d dx e-λx = λe-λx . (d) Set the CDF F X ( x ) = 1-e-λx equal to 0.5. Use the natural log function to solve for x . 1...
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This note was uploaded on 02/06/2012 for the course STAT 225 taught by Professor Martin during the Spring '08 term at Purdue University-West Lafayette.

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