Life Data Analysis with a Joint Probability Density Function

When assessing product life, the survival analysis is generally conducted in a time or usage domain. In certain instances, it is beneficial to investigate the impact of joint variables on product reliability, and in this case, the joint distribution of usage and time are considered. While it is simple to analyze unconditional probability density functions independently, the problem of statistical independence with random variables of usage and time arises. Usage is not independent of time, since usage contains information about time and thus, the joint probability density function then cannot be the product of both marginal probability density functions.