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

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. $$ \begin{array}{l}{\text { a. Find the probability that a bulb will last less than its guaranteed }} \\ {\text { lifetime of } 1000 \text { hours. }} \\ {\text { b. In a batch of light bulbs, what is the expected time until half }} \\ {\text { the light bulbs in the batch fail? }}\end{array} $$

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.
$$
\begin{array}{l}{\text { a. Find the probability that a bulb will last less than its guaranteed }} \\ {\text { lifetime of } 1000 \text { hours. }} \\ {\text { b. In a batch of light bulbs, what is the expected time until half }} \\ {\text { the light bulbs in the batch fail? }}\end{array}
$$

Key Concepts

Probability Density and Cumulative Distribution Functions

The probability density function (pdf) of the exponential distribution is given by f(t) = ? exp(??t) for t ? 0. The cumulative distribution function (CDF), which is used to determine the probability that the observed value is less than or equal to a certain threshold, is F(t) = 1 ? exp(??t). These functions are essential tools for calculating probabilities and quantiles, such as the probability that an event (like failure) occurs within a specified time frame.

Median of the Exponential Distribution

The median of the exponential distribution is the value m for which the cumulative distribution function equals 0.5, meaning there is a 50% chance that the event occurs before time m. It can be derived by solving 1 ? exp(??m) = 0.5, giving m = (ln 2)/?. This measure is a practical indicator in reliability studies to estimate the time by which half of the subjects will have experienced the event.

Parameter Estimation from Mean

For the exponential distribution, the mean is the inverse of the rate parameter (?). When the mean is known, the rate parameter can be calculated as ? = 1 divided by the mean. This connection between the mean and the rate parameter is crucial for forming the probability density function (pdf) and the cumulative distribution function (CDF) of the distribution.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time between events in a Poisson process. It is characterized by a constant hazard rate, meaning that the probability of an event occurring in the next moment is independent of how much time has already passed. This property makes the distribution particularly useful in reliability analysis and in modeling lifetimes of devices or components.

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