Answer to Question #115142 in Statistics and Probability for Warith

Question

The measurement instruments used in the tool room of a company are found [2] to have failure rate of 0.02 per year. What does this mean? What is the
relationship between Reliability and Failure rate?

Answer 1

The reliability function, also called the survivor function or the probability of success, is denoted by R(t). It represents the probability that a brand new component will survive longer than a specified time.

A way to look at the failure behavior in time is to examine the failure rate. Failure rate is the time rate of change of the probability of failure. Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. 

It can be computed by finding the area under the pdf to the right of a specified time, or:

$$R(t)=\displaystyle\int_{-\infin}^tf(s)ds$$

Conversely, if the reliability function is known, the pdf can be obtained as:

$$f(t)=-{dR(t)\over dt}$$

The cumulative distribution function (CDF), also called the unreliability function or the probability of failure, is denoted by $F(t).$

The reliability function and the unreliability function satisfy the following equation:

$$R(t)+F(t)=1$$

The failure rate function, also called the instantaneous failure rate is denoted by $\lambda(t).$ It represents the probability of failure per unit time, $t,$ given that the component has already survived to time $t.$ Mathematically, the failure rate function is a conditional form of the pdf, as seen in the following equation:

$$\lambda(t)={f(t)\over R(t)}$$

The Mean Time Between Failures(MTBF, 1/λ)

$$MTBF={1\over \lambda}={Total\ time\over Number\ of\ failures}$$

The measurement instruments used in the tool room of a company are found [2] to have failure rate of 0.02 per year. 

$$\lambda(t)={f(t)\over R(t)}={0.02\over 365\times24}={1\over 438000}\ per\ hour$$

 Suppose 50 devices are tested for 8760 hours (1 year). During the test 1 failure occur.