The blade and the bearings are important parts of a lathe....

The blade and the bearings are important parts of a lathe. The lathe can operate only when both of them work properly. The lifetime of the blade is exponentially distributed with the mean three years; the lifetime of the bearings is also exponentially distributed with the mean four years. Assume that each lifetime is independent. (a) What is the probability that the lathe will operate for at least five years? (b) The lifetime of the lathe exceeds what time with $95 \%$ probability?

The blade and the bearings are important parts of a lathe. The lathe can operate only when both of them work properly. The lifetime of the blade is exponentially distributed with the mean three years; the lifetime of the bearings is also exponentially distributed with the mean four years. Assume that each lifetime is independent.
(a) What is the probability that the lathe will operate for at least five years?
(b) The lifetime of the lathe exceeds what time with $95 \%$ probability?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly used to model the time until an event, such as a component failure, occurs. It is characterized by its constant hazard rate and memoryless property, making it a suitable model for independent, exponentially distributed lifetimes.

Minimum of Exponential Random Variables

When considering two independent exponentially distributed random variables, the minimum of these variables is also exponentially distributed with a rate parameter equal to the sum of the individual rates. This concept is particularly useful in reliability analysis of systems where the overall operation depends on the performance of the component with the shortest lifetime.

Reliability of Series Systems

In reliability theory, a series system is one where the system fails as soon as any one of its components fails. The total system reliability is therefore determined by the component with the shortest lifetime. For systems with exponentially distributed component lifetimes, the overall system lifetime is the minimum of these lifetimes.

Quantiles of the Exponential Distribution

Quantiles of the exponential distribution are used to determine the time by which a certain proportion of the population of devices will have failed. Calculating these allows one to find, for example, the time by which there is a specified probability (such as 95%) that the device or system will still be operational.

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