Lifetime of light bulbs A manufacturer of light bulbs finds...

Name: Problem 31, Chapter 8: Thomas Calculus
Uploaded: 2020-02-06T21:33:11-08:00
Duration: 2 min 19 s
Channel: Clarissa Noh
Description: Lifetime of light bulbs A manufacturer of light bulbs finds that the mean lifetime of a bulb is 1200 hours. Assume the life of a bulb is exponentially distributed. a. Find the probability that a bulb will last less than its guaranteed lifetime of 1000 hours. b. In a batch of light bulbs, what is the expected time until half the light bulbs in the batch fail?

Lifetime of light bulbs A manufacturer of light bulbs finds that the mean lifetime of a bulb is 1200 hours. Assume the life of a bulb is exponentially distributed. a. Find the probability that a bulb will last less than its guaranteed lifetime of 1000 hours. b. In a batch of light bulbs, what is the expected time until half the light bulbs in the batch fail?

Lifetime of light bulbs A manufacturer of light bulbs finds that the mean lifetime of a bulb is 1200 hours. Assume the life of a bulb is exponentially distributed.
a. Find the probability that a bulb will last less than its guaranteed lifetime of 1000 hours.
b. In a batch of light bulbs, what is the expected time until half the light bulbs in the batch fail?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly used to model the time between events, such as the failure of electronic components or light bulbs. It is characterized by a constant rate of occurrence, which means the probability of an event occurring in the next interval is independent of how much time has already elapsed.

Rate Parameter and Mean

In the context of the exponential distribution, the rate parameter (often denoted by ?) is the reciprocal of the mean. The mean lifetime of a component is directly related to the rate at which failures occur, and knowing one allows you to calculate the other. This relationship is fundamental in reliability studies, as it simplifies the process of deriving other properties of the distribution.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for the exponential distribution provides the probability that the time until an event occurs is less than or equal to a specified value. It is expressed as F(t) = 1 - e^(-?t), where t represents time. This function is essential for calculating probabilities related to lifetimes, such as the chance that a product fails before its guaranteed lifetime.

Median Lifetime

The median of an exponential distribution is the time by which there is a 50% probability that the event has occurred, commonly used in reliability analysis to determine the time by which half the components have failed. It is computed by solving F(t) = 0.5, leading to t = ln(2)/?, and provides a useful measure of central tendency for skewed lifetime data.

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