# hw07 - (b). Problem 3. The length X of a side of a cube has...

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ORIE 360/560 Fall 2007 Assignment 7 Due Tuesday, Oct. 23 at 2:00 pm Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. The number of minutes airline ﬂights are behind schedule is assumed to follow a gamma distribution with shape parameter α = 2 and rate parameter λ = 1 / 4. (a) Find the probability that a ﬂight is at least than 20 minutes late. Hint: recall that if α > 0 is an integer, then Γ( α ) = ( α - 1)! . (b) Use Markov’s inequality to ﬁnd a bound on the probability from part (a). (c) Use Chebyshev’s inequality to ﬁnd a bound on the probability from part (a). Problem 2. A fair coin is tossed 1 , 000 times. Let H be the number of heads tossed. (a) Find a and b such that aH + b is approximately distributed as N (0 , 1). (b) Using your answer to part a , approximate the probability that H < 350. (You will need to use Excel or some other software package that can compute the normal cdf.) (c) As in Problem 1, use Markov’s and Chebyshev’s inequalities to ﬁnd bounds on the probability from part
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Unformatted text preview: (b). Problem 3. The length X of a side of a cube has pdf f X ( x ) = x-2 , for 1 x . Find the pdf of the surface area of the cube. Problem 4. Suppose you can generate iid uniform (0,1) random variables U 1 ,U 2 ,... . Use them to generate n independent Gamma( m, ) random variables, where n and m are positive integers and > 0. Problem 5. A random vector ( X,Y ) has joint pdf f X,Y ( x,y ) = ( e-y if 0 < x < y, otherwise . Find the joint pdf of Z = Y-X and V = X . Are Z and V independent? The following problem is optional for students enrolled in 360, required for students enrolled in 560. Problem 6. A point in the plane is generated at random according to the following scheme. A radius R is generated from the 2 2 distribution. Independently, an angle is generated from the U (0 , 2 ) distribution. Find the joint distribution of X = R cos and Y = R sin ....
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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 University (Engineering School).

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