Quiz4Solutions

# Quiz4Solutions - 1 and look up the answer in a table P...

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STA 3032 - Section 7661 - Fall 2011 - Quiz 4 Please write your name and UF ID number and answer all questions on the paper that is provided to you. Do not write on the quiz sheet. Anything you write on the quiz sheet will not be graded. Show all work. Continuous Uniform f ( x ; a,b ) = 1 b - a ,x [ a,b ] Normal f ( x ; μ,σ 2 ) = 1 2 πσ 2 exp ( - 1 2 σ 2 ( x - μ ) 2 ) ,x ( -∞ , ) Gamma f ( x ; α,β ) = 1 Γ( α ) β α x α - 1 exp( - x/β ) ,x [0 , ) Exponential f ( x ; β ) = 1 β exp( - x/β ) ,x [0 , ) Beta f ( x ; α,β ) = Γ( α + β ) Γ( α )Γ( β ) x α - 1 (1 - x ) β - 1 ,x [0 , 1] 1. Let f be deﬁned as the following: f ( x ) = k x, x [0 , ) 0 , elsewhere Find the value of k that makes f a valid pdf. (6 points). Answer: Recall that f is a pdf if if is non-negative for all x and integrates to 1. 1 = Z 1 0 f ( x ) dx = Z 1 0 k xdx = k ± x 3 / 2 3 / 2 ² ² ² 1 0 ³ 1 = k/ (3 / 2) k = 3 / 2 . 2. Let X N (5 , 100) . Find P ( X > - 1 . 12) . (7 points) Answer: Recall that the pdf for the normal distribution cannot be easily integrated analytically. We convert X to Z N (0 ,

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Unformatted text preview: 1) and look up the answer in a table. P ( X >-1 . 12) = P ´ X-5 10 >-1 . 12-5 10 µ = P ( Z >-. 612) = P ( Z < . 612) ≈ P ( Z < . 61) = 0 . 7291 . We can say P ( Z >-. 612) = P ( Z < . 612) since the pdf of the Z curve is symmetric about zero. 3. The lifetime of a particular integrated circuit has an exponential distribution with mean 2 years. Find the probability that the circuit lasts longer than three years. (7 points) Answer: Recall that the Exponential distribution has a mean equal to β . Then in our problem β = 2 . P ( X > 3) = Z ∞ 3 1 2 e-x/ 2 dx = 1 2 ± e-x/ 2-1 / 2 ² ² ² ∞ 3 ³ =-± lim L-> ∞ e-L/ 2-e-3 / 2 ³ =-´-e-3 / 2 µ = e-3 / 2 = 0 . 2231302 . 2...
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## This note was uploaded on 12/13/2011 for the course STA 3032 taught by Professor Kyung during the Fall '08 term at University of Florida.

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Quiz4Solutions - 1 and look up the answer in a table P...

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