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HW4_Problem_Set

# HW4_Problem_Set - wise she will go away alone If Julia...

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ORIE 3500/5500 Fall Term 2008 Assignment 4 Note that the due date is Monday, October 6 , at 1:00 pm you need to mention your section number on the top of your answer script. 1. [2 + 3 + 2 + 3 + 5 = 15 pts] A fair 6-sided die is rolled n times. For i = 1 , 2 , . . . , 6 define the random variables X i as follows. X i = 1 if i does not occur in the rolls 0 otherwise . (a) Find E ( X i ) , 1 i 6. (b) Find E ( X i X j ) , 1 i, j 6 , i = j . (c) If N is the number of integers from { 1 , 2 , . . . , 6 } that do not occur in any of the n rolls, then express N in terms of X i s’. (d) Find E ( N ). (e) Find var ( N ). 2. [3 + 3 + 4 = 10 pts] Suppose X is a continuous random variable with density f ( x ) = ce - 5 | x | , x R . (a) Find c . (b) Find P ( X ( - 1 , 1)). (c) Assuming that E ( X 2 ) < , find var ( X ). 3. [5 + 5 = 10 pts] Julia is waiting for Tim in a coffee-shop. Tim will arrive after H hour, where H has density h ( x ) = 2 x, 0 x 1 . If Tim comes within 30 minutes, then Julia will go with him and other-

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Unformatted text preview: wise she will go away alone. If Julia waits for T hour in the coﬀee-shop, ﬁnd (a) the cdf of T and (b) E ( T ). 1 4. [2 × 5 = 10 pts] Let X be a random variable with distribution function F ( x ) = (1 / 4) e x-1 if-∞ < x < 1 1 / 2 if 1 ≤ x < 2 x/ 4 if 2 ≤ x < 3 1 if x ≥ 3 . (a) Is X discrete, continuous or mixed? (b) Find P (0 < X ≤ 1 . 5). (c) Find P ( X = 3). (d) Find P (1 ≤ X < 3). (e) Find P (-1 ≤ X ≤ 1). 5. [2 + 3 = 5 pts] Suppose X has a normal distribution with mean 2 and variance 4. If Y = ( X-2) / 2, (a) ﬁnd the PDF of Y . (b) Find P (-2 ≤ X ≤ 2)? 2...
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HW4_Problem_Set - wise she will go away alone If Julia...

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