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Probability Density Function (PDF)
For a continuous function, the probability density function (PDF) is the probability that the variate has the value x. Since for continuous distributions the probability at a single point is zero, this is often expressed in terms of an integral between two points.
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) is the probability that the variate takes on a value less than or equal to x:
For a continuous distribution, this can be expressed as
Survival Function
The survival function (or reliability function) is the probability that the variate takes on a value greater than x. This function is often used in reliability and related fields to denote the probability a unit survives beyond time t.
Hazard Function
The hazard function (also known as the failure rate, or hazard rate) is the ratio of the probability density function to the survival function:
The hazard function is used in reliability applications to describe the instantaneous failure rate at any point in time.
If the hazard function is constant, then the failures occur with equal frequency during any equal period of time. The exponential failure distribution has a constant hazard rate. For other distributions, the hazard function is not constant, so the failure rate varies with time.
