(Solution) - A light fixture contains five lightbulbs The lifetime of each -(2025 Original AI-Free Solution)

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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 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 as exponentially distributed with parameter ?, what would be the distribution of T?