The probability density function of the time to failure of...

The probability density function of the time to failure of an electronic component in a copier (in hours) is $f(x)=$ $e^{-x / 1000} / 1000$ for $x>0 .$ Detcrmine the probability that (a) A component lasts more than 3000 hours before failure. (b) A component fails in the interval from 1000 to 2000 hours. (c) A component fails before 1000 hours. (d) Determine the number of hours at which $10 \%$ of all components have failed.

The probability density function of the time to failure of an electronic component in a copier (in hours) is $f(x)=$ $e^{-x / 1000} / 1000$ for $x>0 .$ Detcrmine the probability that
(a) A component lasts more than 3000 hours before failure.
(b) A component fails in the interval from 1000 to 2000 hours.
(c) A component fails before 1000 hours.
(d) Determine the number of hours at which $10 \%$ of all components have failed.

Key Concepts

Survival Function

The survival function, also known as the reliability function, is the complement of the CDF. It gives the probability that a component lasts longer than a given time. This function is essential for assessing the durability and reliability of components in reliability engineering.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) represents the probability that the random variable is less than or equal to a specified value. It is calculated by integrating the PDF from zero up to the value of interest, and is key in determining the probabilities associated with various time intervals.

Percentile/Quantile Calculation

Percentile or quantile calculations involve finding the point in the probability distribution at which a given percentage of observations fall below. In reliability analysis, this technique is used to determine the time by which a certain percentage of items are expected to fail, providing important information for maintenance planning and quality control.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is widely used to model the time between events, such as the failure of a component. It has the notable property of a constant hazard rate, meaning that the probability of failure in the next instant is independent of how long the component has already functioned.

Probability Density Function (PDF)

The probability density function (PDF) specifies the likelihood of the random variable taking on a specific value. For the exponential distribution, the PDF exhibits an exponential decay, illustrating that the probability of a failure decreases as time increases.

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