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# HW4 - ﬁnd the density of the time of the ﬁrst...

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OR 3500/5500, Summer’09, Chen Homework 4 Due on Monday, June 1, 3pm. Problem 1 Suppose that density of X is given by f ( x ) = a + bx 2 , 0 x 1 If E(X)=c, compute a and b in terms of c . Problem 2 Let X 1 , X 2 , . . . , X n be independent Uniform(0,1) random variables ( ie , each one of them have density given by f ( x ) = 1 , 0 x 1). (a) Compute the cdf of Y := min( X 1 , . . . , X n ). (b) Use (a) to compute the density of Y . (c) Find E ( Y ). Problem 3 Consider an experiment having 3 possible outcomes that occur with probabilities p 1 , p 2 and p 3 respectively. Suppose n independent repetitions of the experiment are made and let X i denote the number of times the i th outcome occurs. (a) What is the pmf of X 1 + X 2 ? (b) Find P [ X 2 = j | X 1 + X 2 = i ], j=0,1,. . . ,i. Problem 4 In a factory, there is a certain equipment which needs to be replaced immediately after it goes bad. Let X 1 and X 2 denote the first two times (in days) when the equipment is replaced. The joint density of X 1 and X 2 is given by f X 1 ,X 2 ( x 1 , x 2 ) = e - x 2 , 0 x 1 x 2 (a) Compute the density of the lifetime of the equipment ( ie marginal of X 1 ).

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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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HW4 - ﬁnd the density of the time of the ﬁrst...

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