The definition of reliability is the probability that a system will perform its intended function for a specified period of time under stated condition. Note that we are inferring the probability after conditioning on the performance for certain amount of time.

MTBF and hazard function are two popular measures. Inverse of the failure rate is the mean time between failure and the rate (lambda) is the focus for Bayesian update as it allows the conjugacy between gamma and poisson. To understand the following think let’s think that r = 1. The failure event with the rate of lambda (lambda times for unit time) would happen lamda * t for time t leading to poisson pdf with rate lambda * t.