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Unformatted text preview: ﬁnd the density of the time of the ﬁrst replacement. Problem 5 Let X be Uniform(0,1) (deﬁned in problem 2.). Deﬁne Y = ‰ 1 , if X ≤ 1 / 3 2 , if X > 1 / 3 (a) Compute E ( Y ) by ﬁrst deriving its pmf. (b) Now verify your answer by computing E ( Y ) directly, using the formula for the expectation of a function of a random variable. The following problem is optional for 3500 students, mandatory for students enrolled in 5500 1 OR 3500/5500, Summer’09, Chen Problem 6 Let ( X,Y,Z ) be jointly distributed with joint pdf f X,Y,Z ( x,y,z ) = Ce( x + y + z ) , < x < y < z. Find C . Find the marginal joint pdf of ( X,Y ) and compute P [ X + Y ≤ 2]. What are the marginal pdfs of X and Y . Show that X,Y and Z are not independent and ﬁnd the conditional density of X given Y . 2...
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 Summer '08
 WEBER
 Probability theory, probability density function, Cumulative distribution function, 15 days

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