Question

The life length y of a component used in a complex electronic system is known to have an exponential density with a mean of 50 hours. the component is replaced at failure or at age 100 hours, whichever comes first.

a. find the distribution function for x, the length of time the component is in use.
b. find e(x).

Answer 1

The student's question relates to finding the distribution function and expected value (E(x)) for a component's usage time, given an average lifespan and replacement policy. This involves applying the properties of exponential distribution and adjusting for the maximum usage limit.

Step-by-step explanation:

The question deals with the exponential distribution of a component's life length, which has an average life span of 50 hours and is subject to replacement at the earlier occurrence of its failure or reaching 100 hours of usage.

To address part a of the question, the distribution function for the length of time the component is in use x, needs to be determined for x ranging from 0 to 100 hours, considering the survival function of exponential distribution and adjusting for the cut-off at 100 hours.

For part b, the calculation of the expected value E(x) involves integrating the density function of x over its range. The expected value would be influenced by the fact that the distribution is truncated at 100 hours, since components are replaced at this time irrespective of whether they've failed or not.