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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 ) = ± cx3 , 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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 Fall '08
 WEBER

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