Given a sequence of data points
,
its cumulative distribution function
is defined by
That is, is the relative proportion of the data points taking value
less than or equal to
.
Given the data points
, compute
where
is the corresponding cumulative distribution function.
There are a total of sixteen data points of which nine have a value less
than or equal to four. Thus,
.
One may regard the cumulative distribution function as
describing the probability that a randomly chosen data point will
have value less than or equal to
.
If is the correponding random variable, one often writes
From we may compute other probabilities. For instance,
the probability of obtaining a value greater than
but less than
or equal to
is
We may wish to express the probability that a numerical value of a particular experiment lie with a certain range even though infinitely many such values are possible.
A cumulative distribution function (in general) is a function
defined for all real numbers for which
We write for the corresponding random variable and treat
as
expressing
.
If the cumulative distribution function (for the random variable
)
is differentiable and have derivative
, then we say that
is
the probability density function for
.
For numbers we have
Conversely, any function satisfying the above properties is a probability density.
The function
is a probability density (for the random variable ).
Compute
.
We know