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While the whole population of a group has certain characteristics, we can typically never measure all of them. In many cases, the population distribution is described by an idealized, continuous distribution function.
In the analysis of measured data, in contrast, we have to confine ourselves to investigate a (hopefully representative) sample of this group, and estimate the properties of the population from this sample.
A continuous distribution function describes the distribution of a population, and can be represented in several equivalent ways:
The PDF, or density of a continuous random variable, is a function that describes the relative likelihood for a random variable to take on a given value . In the mathematical fields of probability and statistics, a random variate x is a particular outcome of a random variable X: the random variates which are other outcomes of the same random variable might have different values.
Since the likelihood to find any given value cannot be less than zero, and since the variable has to have some value, the PDF has the following properties:
Probability Density Function (PDF) of a value x. The integral over the PDF between a and b gives the likelihood of finding the value of x in that range.
The probability to find a value between and is given by the integral over the PDF in that range (see Fig. [fig:PDF]), and the Cumulative Distribution Function tells you for each value which percentage of the data has a lower value (see Figure below). Together, this gives us
Probability Density Function (left) and Cumulative distribution function (right) of a normal distribution.
The Figure Utility functions for continuous distributions, here for the normal distribution. shows a number of functions are commonly used to select appropriate points a distribution function:
Probability density function (PDF): note that to obtain the probability for the variable appearing in a certain interval, you have to integrate the PDF over that range.
Example: What is the chance that a man is between 160 and 165 cm tall?
Cumulative distribution function (CDF): gives the probability of obtaining a value smaller than the given value.
Example: What is the chance that a man is less than 165 cm tall?