Template:ProbabilityDensitynCumulativeDistributionFunctions

The Probability Density and Cumulative Distribution Functions
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 $$X$$ is a continuous random variable, then the probability density function, $$pdf$$, of $$X$$, is a function $$f(x)$$ such that for two numbers, $$a$$ and $$b$$ with $$a\le b$$:
 * $$P(a \le X \le b)=\int_a^b f(x)dx$$  and  $$f(x)\ge 0 $$ 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 $$a$$ to $$b$$. The cumulative distribution function, $$cdf$$, is a function $$F(x)$$ of a random variable, $$X$$, and is defined for a number $$x$$ by:




 * $$F(x)=P(X\le x)=\int_0^x f(s)ds $$

That is, for a given value $$x$$, $$F(x)$$ is the probability that the observed value of $$X$$ will be at most $$x$$. Note that the limits of integration depend on the domain of $$f(x)$$. For example, for all the distributions considered in this reference, this domain would be $$[0,+\infty]$$,  $$[-\infty ,+\infty]$$ or $$[\gamma ,+\infty]$$. In the case of $$[\gamma ,+\infty ]$$, we use the constant $$\gamma $$ 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.

Mathematical Relationship Between the $$pdf$$ and $$cdf$$

The mathematical relationship between the $$pdf$$ and $$cdf$$ is given by:


 * $$F(x)=\int_{-\infty }^x f(s)ds$$

Conversely:


 * $$f(x)=\frac{d(F(x))}{dx}$$

In plain English, the value of the $$cdf$$ at $$x$$ is the area under the probability density function up to $$x$$, if so chosen. It should also be pointed out that the total area under the  $$pdf$$ is always equal to 1, or mathematically:


 * $$\int_{-\infty }^{\infty }f(x)dx=1$$



An example of a probability density function is the well-known normal distribution, whose $$pdf$$ is given by:


 * $$f(t)={\frac{1}{\sigma \sqrt{2\pi }}}{e^{-\frac{1}{2}}(\frac{t-\mu}{\sigma})^2}$$

where $$\mu $$ is the mean and $$\sigma$$ is the standard deviation. The normal distribution is a two-parameter distribution, i.e. with two parameters $$\mu $$ and $$\sigma $$. Another two-parameter distribution is the lognormal distribution, whose $$pdf$$  is given by:


 * $$f(t)=\frac{1}{t\cdot {{\sigma }^{\prime }}\sqrt{2\pi }}{e}^{-\tfrac{1}{2}(\tfrac{t^{\prime}-{\mu^{\prime}}}{\sigma^{\prime}})^2}$$

where $$ t^{\prime}$$ is the natural logarithm of the times-to-failure, $$\mu^{\prime}$$ is the mean of the natural logarithms of the times-to-failure and $$\sigma^{\prime}$$ is the standard deviation of the natural logarithms of the times-to-failure, $$ t^{\prime }$$.