Last edit: 26/02/2024
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:
This probability is shown in figure 1.6 {1.4.2.2}.
The Probability Density Function is also called Failure Density or also Life Distribution.
The distribution of a continuous variable can be described by the Cumulative Distribution Function as well. That gives the probability that the random variable will assume a value smaller or equal to x. Its expression is :
For -∞< x <+∞.
F(x) is a non-decreasing function : F(-∞) = 0 et 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 P( a ≤ X ≤ b ) 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 :
