A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let T be the time of the first bulb replacement. Let Xi, i = 1, …, 5, be the lifetimes of the five bulbs. Assume the lifetimes of the bulbs are independent. a) Find P(X1 > 100). b) Find P(X1 > 100 and X2 > 100 and ⋯ and X5 > 100). c) Explain why the event T > 100 is the same as {X1 > 100 and X2 > 100 and ⋯ and X5 > 100}. d) Find P(T ≤ 100). e) Let t be any positive number. Find P(T ≤ t), which is the cumulative distribution function of T. f) Does T have an exponential distribution? g) Find the mean of T. h) If there were n lightbulbs, and the lifetime of each was exponentially distributed with parameter λ, what would be the distribution of T?
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A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let T be the time of the first bulb replacement. Let Xi, i = 1, …, 5, be the lifetimes of the five bulbs. Assume the lifetimes of the bulbs are independent. a) Find P(X1 > 100). b) Find P(X1 > 100 and X2 > 100 and ⋯ and X5 > 100). c) Explain why the
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