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 :