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Unformatted text preview: IE 111 Spring 2009 Homework #7 Solutions Question 1. Consider the following probability density function (PDF): f X (x) = kx 2 for 0 ≤ x ≤ 1 f X (x) = 0 otherwise a) Find k so that we have a valid probability density function. 3 3 3 1 1 1 3 2 = = = = ∫ k thus k x k dx x k b) Find P(X ≤ 1). Since the domain is [0,1], P(X ≤ 1) = 1 c) Find E(X). 4 3 4 3 3 1 1 4 2 = = ∫ x dx xx d) Find b so that P(X ≤ b) = 0.9. 0.965489 9 . 9 . 1 3 3 3 ) ( 3 3 3 3 2 = = ⇒ = < < = = = ∫ x x x x x dx x x F x x Question 2. Consider the following probability density function (PDF): f X (x) = kx for 0 ≤ x ≤ 1 f X (x) = k for 1 ≤ x ≤ 2 f X (x) = 0 otherwise a) Find k so that we have a valid probability density function. AREA = (k/2) + k therefore k = 2/3 b) Find P(X ≤ 1). P(X ≤ 1) = 1/3 c) Find E(X). 9 11 6 2 8 9 2 6 2 9 2 3 2 3 2 ) ( 2 1 2 1 3 1 2 1 = + = + = + = ∫ ∫ x x xdx xxdx X E d) Find P(X>1.5). = (0.5)(2/3) = 1/3 Question 3. Visitors arrive to a web site according to a Poisson process. On average we get one arrival every 15 seconds. a) What is the probability that we receive 16 or more visits in a 5 minute time period? λ =(1 per 15 sec)(20 15 sec periods in 5 minutes) = 20 P(X≥16) = 1P(X≤15) = 1 0.156513 = 0.843487 b) Starting at 12 noon, what is the probability that the first visitor arrives after 12:01? P(First after 12:01) = P(no arrivals in 12:00 to 12:01) λ =(1)(4) = 4 P(X=0) = e4 = 0.018316 Question 4 Dust particles land on my newly painted wall at a rate of 2.5 per hour. The wall is 8 feet by 12 feet. As long as the paint is wet, the dust particles will stick. It takes 8 hours for the paint to dry. a) Find the probability that exactly 20 particles stick to my wall. λ =(2.5)(8) = 20 P(X=20) = 0.088835 b) Find the probability that 20 or fewer particles stick to my wall. P(X≤20) = 0.559093 c) Find the probability that 10 or more particles stick to my wall. P(X≥10) = 1P(X≤9) = 1 0.004995 = 0.995005 d) I decided to paint another wall which is 8 feet by 6 feet. Let X be the total number of dust particles that stick (over both walls). Find E(X). State any necessary assumptions. State any necessary assumptions....
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This note was uploaded on 01/04/2010 for the course STATS 1100 taught by Professor Rodriguez during the Spring '08 term at Pittsburgh.
 Spring '08
 Rodriguez
 Statistics, Probability

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