Failure and Reliability 
The failure of a unit in a system means the termination of the unit's ability to perform the required function. Suppose that the time to failure has the probability density function . The failure distribution function is the probability of an item failing in the time interval The reliability function or survivor function is the probability of a unit which does not fail in the time interval The probability of the same unit that will fail in the time interval is the conditional probability of not failing before and failing in , i.e., The failure rate of the unit is: The mean time to failure (MTTF) of a unit is: If the unit is in a system which can be repaired or replaced, the information of the mean time between failure (MTBF) makes more sense. When the time required to repair or replace is much much shorter than MTTF, MTBF is about the same as MTTF. When the repair/replace time cannot be neglected, the mean time to repair (MTTR) has to be taken into consideration. 
Relationship between Failure and Reliability  

Reliability Described by Exponential Distribution  
The exponential distribution is one of the most common distributions used to describe the reliability of a unit in a system. Suppose that the reliability of a unit exponentially decays through time.
where is a positive constant. All the reliability and failure functions can be summarized as follows:
