(a) A type of lightbulb is labeled as having an average...

(a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean $\mu=1000 .$ Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours. (b) What is the median lifetime of these lightbulbs?

(a) A type of lightbulb is labeled as having an average lifetime
of 1000 hours. It's reasonable to model the probability of
failure of these bulbs by an exponential density function
with mean $\mu=1000 .$ Use this model to find the probability
that a bulb
(i) fails within the first 200 hours,
(ii) burns for more than 800 hours.
(b) What is the median lifetime of these lightbulbs?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Median of an Exponential Distribution

The median of an exponential distribution is the point at which the CDF equals 0.5, meaning that half of the observations are expected to be below this value and half above. It is calculated using the formula median = ? ln(2), where ? is the mean of the distribution, providing a measure of central tendency that is less influenced by the tail behavior than the mean.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for an exponential distribution expresses the probability that the random variable is less than or equal to a given value. It is defined as F(t) = 1 - exp(-t/?) for t ? 0, which allows one to compute probabilities over intervals, such as the likelihood of an event occurring within a certain time period or the chance to survive beyond a certain threshold.

Exponential Distribution

The exponential distribution is a continuous probability model often used to describe the time between events in processes that occur continuously and independently at a constant average rate. It is characterized by a single parameter, the mean (or equivalently the rate parameter), and exhibits the memoryless property, meaning that the probability of an event occurring in the future is independent of how much time has already elapsed.

Please give Ace some feedback