{"id":50173,"date":"2025-11-14T15:22:45","date_gmt":"2025-11-14T14:22:45","guid":{"rendered":"https:\/\/www.gt-engineering.it\/?post_type=pubblicazione&p=50173"},"modified":"2026-02-09T13:02:13","modified_gmt":"2026-02-09T12:02:13","slug":"p4-1-functional-safety-pfd-calculation-first-part","status":"publish","type":"pubblicazione","link":"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p4-1-functional-safety-pfd-calculation-first-part\/","title":{"rendered":"P4: Functional Safety – PFD calculation"},"content":{"rendered":"\n\n \n \n \n
\n\n \n \n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n
\n \n
\n \n \n

\n Introduction <\/h2>\n <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

A continuous probability distribution is indicated with f(x)<\/em> and is usually called Probability Density Function (PDF). <\/strong>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.<\/p>\n

The main property of a PDF is that:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

The probability that x<\/em> assumes values between a and b is evaluated as the following integral of the probability density function:<\/strong><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

For\u00a0 – \u221e< x <+ \u221e.<\/p>\n

F(x) is a non-decreasing function:F(-\u221e)=0 and F (+\u221e)=1 , thus:<\/p>\n

 <\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

The derivative of the cumulative distribution function is the probability density function<\/strong> (or failure density) of the random variable X<\/em>:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

The relationship between the Cumulative distribution function F(x)<\/strong> and the Probability density function f(x)<\/strong> is in figure 1.8 {1.4.3.1}.<\/p>\n

These definitions for F(x) allow to express \u00a0as follows:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

Since we reason in terms of time<\/strong> and time is a positive random variable, the Cumulative Distribution Function<\/strong> can be written in the following way:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

and<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n
\n \n
\n \n \n

\n The Reliability Function R(t) <\/h2>\n <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

R(t)<\/em> is the probability that no failure of item occurs in the interval (0 t].<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

In other terms, R(t) is the probability that an item will operate \u201cfailure-free\u201d in time interval (0, t], while the failure will occur in (t, . Known the probability density function f(x), we have:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

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:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

The density function f(t) can now be expressed as:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n
\n \n
\n \n \n

\n The Failure Rate \u03bb <\/h2>\n <\/div>\n <\/div>\n \n \n
\n \n
\n

The failure rate is the basis of the Functional Safety theory.<\/p>\n

[IEC 61508-4] 3.6 Fault, failure and error<\/em><\/strong><\/p>\n

3.6.16 Failure rate<\/strong>. Reliability parameter \u03bb(t) of an entity (single components or systems) such that \u03bb(t)dt is the probability of failure of this entity within [t, t+dt] provided that it has not failed during [0, t].<\/em><\/p>\n

Mathematically, \u03bb(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:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

Using the equation 1.4.4, it is possible to obtain:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

Integrating the upper equation in time:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

Failure rates and their uncertainties can be estimated from field feedback, using conventional statistics.<\/p>\n

The most diffuse and widely known model for the failure rate is the \u201cbathtub\u201d curve. In the initial phase<\/strong> of the component lifetime, \u03bb(t) decreases rapidly with time; this fact derives from the existence of a \u201cweak\u201d fraction of the population whose defects cause a failure within a short period of time from the moment they are produced.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

In the period called useful life<\/strong>, \u03bb(t) is approximately constant, in case for example, of electronic components. For electromechanical components, \u03bb(t) is a function of time and, in this interval, it constantly increases.<\/p>\n

The last period is characterised by a wear out<\/strong>, with a rapidly increasing failure rate \u03bb(t) caused by the wearing out, aging and fatigue.<\/p>\n

During the useful life of a component with a constant failure rate<\/strong>, considering as an initial condition that Reliability at time 0 is at a maximum and it is equal to 1, we have:<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

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 \u03bb = constant.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

Table 1.3 {1.5.1} shows a summary of the four functions described so far.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n","protected":false},"excerpt":{"rendered":"

A continuous probability distribution is indicated with f(x) and is usually called Probability Density Function (PDF). <\/p>\n","protected":false},"author":1,"featured_media":41084,"parent":49606,"menu_order":3,"template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-50173","pubblicazione","type-pubblicazione","status-publish","has-post-thumbnail","hentry"],"acf":[],"yoast_head":"\nP4: Functional Safety - PFD calculation - Gt-Engineering<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p4-1-functional-safety-pfd-calculation-first-part\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"P4: Functional Safety - PFD calculation - Gt-Engineering\" \/>\n<meta property=\"og:description\" content=\"A continuous probability distribution is indicated with f(x) and is usually called Probability Density Function (PDF).\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p4-1-functional-safety-pfd-calculation-first-part\/\" \/>\n<meta property=\"og:site_name\" content=\"Gt-Engineering\" \/>\n<meta property=\"article:modified_time\" content=\"2026-02-09T12:02:13+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.gt-engineering.it\/wp-content\/uploads\/2023\/02\/normativa-atex-768x512.jpg.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"768\" \/>\n\t<meta property=\"og:image:height\" content=\"512\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p4-1-functional-safety-pfd-calculation-first-part\/\",\"url\":\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p4-1-functional-safety-pfd-calculation-first-part\/\",\"name\":\"P4: Functional Safety - 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