Question: [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,

1. find the PDF, mean and variance of the time between first and second replacement.
2. Suppose a new bulb starts at the beginning of January. If it works fine for the entire year what is the conditional probability that it was of brand 1?
3. 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?

(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?

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document