HW6_Problem_Set

# HW6_Problem_Set - ﬁne for the entire year what is the...

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ORIE 3500/5500 Fall Term 2008 Assignment 6 Note that the due date is Monday, October 27 , at 1:00 pm you need to mention your section number on the top of your answer script. 1. [3 + 3 + 4 = 10 pts] Suppose X,Y, Z are independent Uniform (0 , 1) random variables. (a) Find P ( X > Y + Z ). (b) Find P ( Y < ZX ). (c) Find the CDF of X + Y and use that to ﬁnd its PDF. 2. [3 + 3 + 4 = 10 pts] If Y Uniform (0 , 1) and X | Y Uniform ( - (1 - Y ) , 1 - Y ). (a) Find the PDF of X . (b) Find the mean and variance of X . (c) Find the conditional mean and variance of Y given X = 2 / 3. 3. [5 + 3 + 3 + 4 = 15 pts] Two brands of electric bulbs are used in a housing complex. The lifetime of the two brands are exponentially distributed with mean 9 months and 12 months respectively. Assume that if a bulb goes out, then it is immediately replaced with a bulb randomly chosen from the stock. If the stock always contains 25% brand 1 and 75% brand 2 bulbs, (a) ﬁnd the PDF, mean and variance of the time between ﬁrst and second replacement. (b) Suppose a new bulb starts at the beginning of January. If it works

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Unformatted text preview: ﬁne for the entire year what is the conditional probability that it was of brand 1? (c) Suppose a new bulb starts at the beginning of January. Given that the bulb works ﬁne for the entire year what is the conditional expectation for the time of next replacement? 1 (d) If new bulbs are given to 500 places of the housing complex, then what is the expected number of replacements and variance of the number of replacements within the next year? 4. [5 pts] Suppose that X has exp( λ ) PDF and Y is the integer part of X . Find the PMF and expected value of Y . Can you recognize the PMF? 5. [2 + 4 + 4 = 10 pts] The random variable X has PDF f X ( x ) = ± cx-3 , if 1 ≤ x ≤ 3 0 otherwise . (a) Find the value of c and P ( X < 2). (b) Find the conditional PDF and mean of X given that X < 2. (c) Let Y = X 2 . Find conditional mean and variance of Y given X < 2. 2...
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HW6_Problem_Set - ﬁne for the entire year what is the...

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