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?
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?
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?
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?
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
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