# Continuous Probability Function

Also known as the **Probability Density Function (PDF)**

The word "density" is used because every value has a probability of zero. There are just an infinite number.

This probability distribution function returns the probability of a random variable when you have a continuous distribution.

The convention is to use an uppercase F to represent this function.

This cannot be described with a table as there are an infinite number of values.

It can only be described by a probability density function.

Lets assume that you have the following table of data.

This table shows the height of the people within a small company, to the nearest centimetre.

The definition of "continuous" data is that it does not represent exact values. They have only been measured to a certain degree of accuracy.

For example the height 138cm given to the nearest centimeter could represent any value in the following interval:

It is possible to plot a frequency distribution chart for continuous data but the information must be grouped into intervals.

The following tables show two different ways of grouping this data. The upper boundary of one interval is always the lower boundary of the next interval.

The width of an interval is the difference between the two boundaries (i.e. upper boundary - lower boundary).

Both of these charts are known as grouped frequency distribution charts.

### Continuous Random Variables

And since the probability of the entire sample space that equals 1 must be divided among such a great number of samples, the probability for any one sample is infinitesimally small.

No point has a probability in a continuous distribution, only regions have a probability.

There is no probability per sample, only a probability per region. For regions do contain a proportion of the entire sample space, but points cannot.

So no equation can be given to specify the probability at a given point.

The probability distribution of a continuous random variable cannot be represented in tabular form since there are an infinite amount of possible random variables.

The probability of any particular value is zero as there are an infinite number of values.

The function which takes an interval and returns its probability is called the probability density function.

The word density is used because a probability cannot be assigned to individual variables but only to an interval.

Conditions:

If f(x) is a probability density function then it must obey two conditions:

that the total probability for all possible values of the continuous random variable X is 1:

that the probability density function can never be negative: f(x) > 0 for all x.

### Examples

For example what are the possible heights of a 16 year old. - there is an infinite number of decimal places

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