Probability Functions

The probability and availability studies involve use of mathematical and statistical models, which are probabilistic in nature. The probability functions important for development of reliability and availability theory are:

Probability Density Function (PDF)

The Probability Density Function (PDF) is the basis for predicting the behaviour of any probabilistic situation such as reliability or availability. Given the PDF, one can compute reliability function and the instantaneous density function f (t), usually referred to as failure density function.

• Cumulative Distribution Function (CDF)

This function defines the probability that a random variable 't' lies between some lower limit (often zero) and upper limit 't'. For a continuous variable such as time to failure, the cumulative probability F (t) is given by the integration of the density function between the limits required.

Equation 1

F (t ) = ∫ f ( x) dx                         

In reliability studies, this function is also known as unreliability function. The cumulative distribution function increases from zero to unity as't' increases from its smallest to largest value.

• Reliability Function

The reliability function R (t), which is defined as the probability of a system or an item to perform or operate its required function without failure under given condition for an intended period of operation and it is mathematically given as

Equation 2

 R (t) = 1 - F (t) = 1 - ∫ f (x) dx = ∫ R dx                          

•  Hazard Rate and Hazard Function

The hazard function (t) or instantaneous failure rate function is a conditional expression that an item in service for time "t" will be in the next instant of time "dt" given that it has not previously failed, survived up to time't'

Equation 3

i.e.    P [the unit will fail in (t, t + dt) ≅ λ (t) dt                             

Where λ (t) is the hazard rate at time 't' and 'P' is the probability of occurrence. The mathematical relationship between failure density function f (t) and hazard rate λ (t) is

Equation 4

 λ (t ) = f (t ) / [1 - F (t )]                                     

The general equation showing the relationship between the reliability function R (t) and hazard rate

Equation 5

   R (t) = exp [- ∫ λ (x) dx]                   

When λ is independent of time as in the case of constant failure rate, expression reduces to

Equation 6

        R (t) = e^{- λ (t)}