A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200

Question:

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 function of T .
f. Does T have an exponential Find the mean of T.
h. If there were n lightbulbs, and the lifetime of each as exponentially distributed with parameter λ, what would be the of T?
Distribution
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