Suppose that you have ten lightbulbs, that the lifetime of...

Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the on the lifetimes, and that each lifetime has an exponential distribution with parameter $\lambda$ . (a) What is the probability that all ten bulbs fail before time $t ?$ (b) What is the probability that exactly $k$ of the ten bulbs fail before time $t ?$ (c) Suppose that nine of the bulbs have lifetimes that are exponentially distributed witl parameter $\lambda$ and that the remaining bulb has a lifetime that is exponentially distribute with parameter $\theta$ (it is made by another manufacturer). What is the probability that exactly five of the ten bulbs fail before time $t$ ?

Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the on the lifetimes, and that each lifetime has an exponential distribution with parameter $\lambda$ .
(a) What is the probability that all ten bulbs fail before time $t ?$
(b) What is the probability that exactly $k$ of the ten bulbs fail before time $t ?$
(c) Suppose that nine of the bulbs have lifetimes that are exponentially distributed witl parameter $\lambda$ and that the remaining bulb has a lifetime that is exponentially distribute with parameter $\theta$ (it is made by another manufacturer). What is the probability that exactly five of the ten bulbs fail before time $t$ ?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes (or failures) in a fixed number of independent trials when each trial has the same probability of success (or failure). In this context, if each lightbulb fails by time t with probability p = 1 ? exp(??t), then the number of bulbs that fail in the given period follows a binomial distribution. This distribution is employed to calculate the probability of exactly k bulbs failing in a set of ten.

Independence of Events

Statistical independence means that the lifetimes of individual lightbulbs do not affect each other. This allows the joint probability of multiple bulbs failing by a certain time to be calculated as the product of their individual failure probabilities. Independence is essential for simplifying problems where multiple exponential random variables are involved.

Heterogeneous Populations

When dealing with a set of items that do not all share the same probability distribution—such as nine bulbs with one parameter (?) and one bulb with another parameter (?)—it becomes necessary to handle the overall probability by considering each subgroup separately. This typically involves calculating probabilities for different scenarios (for example, whether the special bulb fails or not) and then combining these probabilities appropriately to account for the situation, such as finding the probability that exactly five of the ten bulbs fail before time t.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time until an event occurs, such as the failure of a lightbulb. It is characterized by the parameter ? (or ? for a slightly different population), which represents the failure rate. One key property of the exponential distribution is its memorylessness, meaning that the probability of failure in the next time interval is independent of how much time has already elapsed.

Cumulative Distribution Function (CDF)

The CDF for an exponential distribution gives the probability that the lifetime of an item is less than or equal to a given time t. It is expressed as 1 ? exp(??t) for a bulb with parameter ?. This function is central to the problem because it provides the failure probability within a timeframe, which is used for calculating the probability of individual and multiple failures.

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