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MA519_JAN00 - In the future bulb#n will be immediately...

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QUALIFYING EXAMINATION JANUARY 2000 MATH 519 - Prof. Sellke Problems are worth 20 points each. A table of the standard normal distribution is attached. 1. Brand A lightbulbs have exponential lifetimes with mean one year. Brand B lightbulbs have exponential lifetimes with mean two years. Brand C lightbulbs have exponential lifetimes with mean three years. A box contains three A bulbs, two B bulbs, and one C bulb. A bulb is chosen at random and screwed into a socket. If the bulb is still working after three years, what is the probability that it is the C bulb? 2. Roll a die 100 times. Let X be the sum of the even numbers rolled. Let Y be the sum of the odd numbers rolled. Calculate the correlation of X and Y . 3. Suppose that Bulgarian lightbulbs have independent lifetimes that are uniformly distributed between 0 years and 1 year. Bulb #1 is screwed into a socket today.
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Unformatted text preview: In the future, bulb #n will be immediately replaced upon burnout with bulb #(n+1). Calculate the probability that the bulb occupying the socket exactly 10 years from now is bulb #18. 4. Let X be a unit exponential random variable, with density f ( x ) = e-x I { x > } . Let Z be standard normal and independent of X . Let T = X + Z . Given that T = 0 (exactly, or to 10 significant digits, whichever you prefer), find the approximate numerical conditional probability that X > 2. 5. Flies enter a room according to a Poisson process with rate 3 per minute. Each time a fly enters the room, the length of time that it stays is uniformly distributed between 0 minutes and 1 minute. Different “sojourn times”are independent. What is the probability that there are exactly 2 flies in the room at a given time? 1...
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