hw03 - be tested one at a time (without replacement) until...

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ORIE 360/560 Fall 2007 Assignment 3 Due Tuesday, Sept. 18 at 2:00 pm Problem 1. In a certain engineering class every student receives a grade of 1.0, 2.0, 3.0, or 4.0., A student has a 30% probability of receiving a failing grade of 1.0 and a 70% probability of receiving a passing grade of 2.0 or higher. Among the students who pass, half receive 2.0, 30% receive 3.0, and the rest receive 4.0. Let X be the grade of a randomly selected student. Is X discrete, continuous, or mixed? Draw the cdf of X . Make sure to label your graph! Problem 2. Suppose X is a continuous random variable with density f ( x ) = ( ce - 3 x if 0 < x < 0 otherwise. (a) Find c . (b) Find P [ X > 3]. Problem 3. Let the point ( X,Y ) be uniformly distributed on an equilateral triangle with side length 1 unit. Find the probability that ( X,Y ) is at least 1/2 a unit of distance away from all three vertices of the triangle. Problem 4. A bin of 5 transistors is known to contain exactly 2 that are defective. The transistors are to
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Unformatted text preview: be tested one at a time (without replacement) until the defective one is identified. Let X be the number of tests made until the first defective one is found (including the test that identifies the first defective one). Let Y be the number of additional tests up to and inlcuding the time that the second defective transistor is found. Find the joint pmf of ( X,Y ). The following problems are optional for students enrolled in 360, required for students enrolled in 560. Problem 5. Suppose X is uniformly distributed on the range [0 , 1]. Find the probability that X is irrational, ie, find P [ X / ∈ Q ]. Problem 6. Again, suppose X ∼ U (0 , 1). Let Y n be the random variable given by Y n = 1 n d nX e , where d x e denotes the least integer k such that x ≤ k . (a) Find the distribution of Y n . (b) Find the distribution of Z n := Y n-X ....
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This note was uploaded on 11/12/2008 for the course ORIE 360 taught by Professor Ehrlichman during the Fall '07 term at Cornell.

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