HW3 - if x < x 2 if 0 x < 1 2 3 if 1...

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OR 3500/5500, Summer’09, Chen Homework 3 Due on Wednesday, May 27, 3pm. Problem 1 Suppose that X is a continuous random variable with the following density f ( x ) = 1 / 3 , - 2 x 1 Compute the density of X 2 . Problem 2 A professor asks her student to do a certain experiment and report some mea- surement. The measurement is a number between 0 and 1 with the following pdf: f ( x ) = cx, 0 x < 1 / 2 c (1 - x ) , 1 / 2 x 1 With probability 1 / 2, the student actually does the experiment and reports the true measurement, and with probability 1 / 2 he feels too lazy to do the experiment and just reports 0 . 5. (a) Find the value of c which makes the given function a legitimate pdf. (b) If Y denotes the measurement the student reports (with or without doing the experiment), compute the cdf of Y (c) What is the type of Y - discrete, continuous or mixed (neither discrete nor continuous)? Problem 3 The distribution function (cdf) of a random variable X is given by F X ( x ) =
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Unformatted text preview: if x &lt; x 2 if 0 x &lt; 1 2 3 if 1 x &lt; 2 11 12 if 2 x &lt; 3 1 if x 3 . ( a ) Plot this cdf. ( b ) What is P ( X &gt; 1 / 2)? ( c ) What is P (2 &lt; X 4)? ( d ) What is P ( X &lt; 3)? ( e ) What is P ( X = 1)? Problem 4 Let X and Y be discrete random variables with joint pmf p XY ( i,N-i ) = 1 N + 1 ,i { , 1 ,...,N } What are the marginal pmfs of X and Y ? 1 OR 3500/5500, Summer09, Chen The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 5 Let X be a number randomly chosen from [0 , 1]. Denote Y n = 1 n d nX e where d x e denotes the smallest integer k so that k x . (a) What is the distribution of Y n ? (b) What is the distribution of Z n := X-Y n ? (c) Compute lim n P ( Y n x ) for 0 x 1. 2...
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HW3 - if x &amp;amp;lt; x 2 if 0 x &amp;amp;lt; 1 2 3 if 1...

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