HW7_Problem_Set

HW7_Problem_Set - ables Deﬁne U = max X 1 X 2 and V = max...

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ORIE 3500/5500 Fall Term 2008 Assignment 7 Note that the due date is Monday, November 3 , at 1:00 pm you need to mention your section number on the top of your answer script. 1. [5 + 5 = 10 pts] Suppose X has PDF f X ( x ) = λe - λx , x 0 , and Y = min( X, 1). Find (a) the transform associated to Y and (b) use it to ﬁnd the mean of Y . 2. [2 + 3 + 2 + 3 = 10 pts] Let X be a discrete random variables that takes nonnegative integer values, and is associated with a transform of the form M X ( s ) = c 1 + 2 e 3 s 3 - e s , where c is some constant. Find (a) p X (3) (b) E ( X ) (c) p X (1) and (d) E ( X | X 6 = 0). 3. [5 + 5 = 10 pts] Suppose X 1 , X 2 , X 3 , X 4 are independent Uniform (0 , 1) random vari-
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Unformatted text preview: ables. Deﬁne U = max { X 1 , X 2 } and V = max { X 3 , X 4 } . Find (a) the PDF of U and V and (b) use convolution rule to ﬁnd the PDF of U + V . 4. [5 + 5 = 10 pts] Suppose X, Y have joint density f X,Y ( x, y ) = 25 e-5 y , ≤ x ≤ y < ∞ . Find (a) the transform for Z = X + 2 Y and (b) use it to ﬁnd var ( Z ). 5. [5 + 5 = 10 pts] Let X and Y be independent with common marginal PDF f ( t ) = 5 e-5 t , t ≥ . Find (a) the PDF of 3 X + 2 Y using convolution rule and (b) use that to ﬁnd its variance. 1...
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