This becomes the instantaneous failure rate or we say instantaneous hazard rate as approaches to zero: A continuous failure rate depends on the existence of a failure distribution, , which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t, where is the failure time. The failure distribution function is the integral of the failure density function, f(t), The hazard function can be defined now as Many probability distributions can be used to model the failure distribution (see List of important probability distributions). A common model is the exponential failure distribution, which is based on the exponential density function. The hazard rate function for this is: Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is "memory-less"). For other distributions, such as a Weibull distribution or a log-normal distribution, the hazard function may not be constant with respect to time. For some such as the deterministic distribution it is monotonic increasing (analogous to "wearing out"), for others such as the Pareto distribution it is monotonic decreasing (analogous to "burning in"), while for many it is not monotonic. Solving the differential equation for , it can be shown that Decreasing failure rate A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. A decreasing failure rate can describe a period of "infant mortality" where earlier failures are eliminated or corrected and corresponds to the situation where λ(t) is a decreasing function. Mixtures of DFR variables are DFR. Mixtures of exponentially distributed random variables are hyperexponentially distributed. Renewal processes For a renewal process with DFR renewal function, inter-renewal times are concave. Brown conjectured the converse, that DFR is also necessary for the inter-renewal times to be concave, however it has been shown that this conjecture holds neither in the discrete case nor in the continuous case. Applications Increasing failure rate is an intuitive concept caused by components wearing out. Decreasing failure rate describes a system which improves with age. Decreasing failure rates have been found in the lifetimes of spacecraft, Baker and Baker commenting that "those spacecraft that last, last on and on." The reliability of aircraft air conditioning systems were individually found to have an exponential distribution, and thus in the pooled population a DFR. Coefficient