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==='''The Probability Density and Cumulative Distribution Functions'''===
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'''Designations'''
 
From probability and statistics, given a continuous random variable,  we denote:
 
:*The probability density  function, ''pdf'', as ''f(x)''.
:*The cumulative distribution  function, ''cdf'', as ''F(x)''.
 
The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable.
 
'''Definitions'''
 
If <math>X</math> is a continuous random variable, then the probability density function, <math>pdf</math>, of <math>X</math>, is a function <math>f(x)</math> such that for two numbers, <math>a</math> and <math>b</math> with <math>a\le b</math>:
 
::<math>P(a \le X \le b)=\int_a^b f(x)dx</math>  and  <math>f(x)\ge 0 </math> for all x.
 
That is, the probability that takes on a value in the interval [a,b] is the area under the density function from <math>a</math> to <math>b</math>.
The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:
 
[[Image:pdf_cdf.png|center|200px|]]
 
::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>
 
That is, for a given value <math>x</math>, <math>F(x)</math> is the probability that the observed value of <math>X</math> will be at most <math>x</math>.
Note that the limits of integration depend on the domain of <math>f(x)</math>. For example, for all the distributions considered in this reference, this domain would be <math>[0,+\infty]</math>,  <math>[-\infty ,+\infty]</math> or <math>[\gamma ,+\infty]</math>. In the case of <math>[\gamma ,+\infty ]</math>, we use the constant <math>\gamma </math> to denote an arbitrary non-zero point (or a location that indicates the starting point for the distribution). The next figure illustrates the relationship between the probability density function and the cumulative distribution function.
<br>
<br>
[[Image:pdf_cdf2.png|200px|center]]
 
'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''
 
The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by:
 
::<math>F(x)=\int_{-\infty }^x f(s)ds</math>
 
Conversely:
 
::<math>f(x)=\frac{d(F(x))}{dx}</math>
 
In plain English, the value of the <math>cdf</math> at <math>x</math> is the area under the probability density function up to <math>x</math>, if so chosen. It should also be pointed out that the '''total area under the ''' <math>pdf</math>  '''is always equal to 1''', or mathematically:
 
::<math>\int_{-\infty }^{\infty }f(x)dx=1</math>
 
 
[[Image:100_percent.png|center|200px|]]
 
 
An example of a probability density function is the well-known normal distribution, whose <math>pdf</math> is given by:
 
::<math>f(t)={\frac{1}{\sigma \sqrt{2\pi }}}{e^{-\frac{1}{2}}(\frac{t-\mu}{\sigma})^2}</math>
 
where <math>\mu </math> is the mean and <math>\sigma</math> is the standard deviation. The normal distribution is a ''two-parameter distribution'', ''i.e.'' with two parameters <math>\mu </math> and <math>\sigma </math>.
Another two-parameter distribution is the lognormal distribution, whose  <math>pdf</math>  is given by:
 
::<math>f(t)=\frac{1}{t\cdot {{\sigma }^{\prime }}\sqrt{2\pi }}{e}^{-\tfrac{1}{2}(\tfrac{t^{\prime}-{\mu^{\prime}}}{\sigma^{\prime}})^2}</math>
 
where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.
 
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