Last edit: 20/08/2025
Introduction
A continuous probability distribution is indicated with f(x) and is usually called Probability Density Function (PDF). It is expressed by an equation and it can be represented as in the Figure 1.5 {1.4.2.1}. The bell curve is just an example of a possible PDF.
The main property of a PDF is that:
The probability that x assumes values between a and b is evaluated as the following integral of the probability density function:
For – ∞< x <+ ∞.
F(x) is a non-decreasing function:F(-∞)=0 and F (+∞)=1 , thus:
The derivative of the cumulative distribution function is the probability density function (or failure density) of the random variable X:
The relationship between the Cumulative distribution function F(x) and the Probability density function f(x) is in figure 1.8 {1.4.3.1}.
These definitions for F(x) allow to express as follows:
Since we reason in terms of time and time is a positive random variable, the Cumulative Distribution Function can be written in the following way:
and
The Reliability Function R(t)
R(t) is the probability that no failure of item occurs in the interval (0 t].
In other terms, R(t) is the probability that an item will operate “failure-free” in time interval (0, t], while the failure will occur in (t, . Known the probability density function f(x), we have:
If the system can be found in two states only, either correct functioning or failure, we can define the function of unreliability F(t) as complementary to R(t), that means:
The density function f(t) can now be expressed as:
The Failure Rate λ
The failure rate is the basis of the Functional Safety theory.
[IEC 61508-4] 3.6 Fault, failure and error
3.6.16 Failure rate. Reliability parameter λ(t) of an entity (single components or systems) such that λ(t)dt is the probability of failure of this entity within [t, t+dt] provided that it has not failed during [0, t].
Mathematically, λ(t) is the conditional probability of failure per unit of time over [t, t+dt]. It is possible to demonstrate that the instantaneous failure rate is:
Using the equation 1.4.4, it is possible to obtain:
Integrating the upper equation in time:
Failure rates and their uncertainties can be estimated from field feedback, using conventional statistics.
The most diffuse and widely known model for the failure rate is the “bathtub” curve. In the initial phase of the component lifetime, λ(t) decreases rapidly with time; this fact derives from the existence of a “weak” fraction of the population whose defects cause a failure within a short period of time from the moment they are produced.
In the period called useful life, λ(t) is approximately constant, in case for example, of electronic components. For electromechanical components, λ(t) is a function of time and, in this interval, it constantly increases.
The last period is characterised by a wear out, with a rapidly increasing failure rate λ(t) caused by the wearing out, aging and fatigue.
During the useful life of a component with a constant failure rate, considering as an initial condition that Reliability at time 0 is at a maximum and it is equal to 1, we have:
The Reliability function R(t) is shown in Figure 1.13a {1.5.2a} and the Probability Density Functions f(t) in Figure 1.13b {1.5.2b}, in the case λ = constant.
Table 1.3 {1.5.1} shows a summary of the four functions described so far.