Examples of distribution functions: the first two are used in binary choice models, the third one is applied in maximum likelihood.
Example 1. Distribution function of a normal variable
The standard normal distribution is defined by its probability density
It is nonnegative and integrates to 1 (the proof of this fact is not elementary). Going from density function to distribution function gives us the distribution function (cdf) of the standard normal:
for all real
.
Example 2. The logistic distribution
Here we go from distribution function to density function.
Consider the function
It's easy to check that it has the three characteristic properties of a distribution function: the limits at the right and left infinities and monotonicity.
1. When ,
goes to 1, so
tends to 1.
2. If ,
goes to
, and
tends to 0.
3. Finally, to check monotonicity, we can use the following sufficient condition: a function is increasing where its derivative is positive. (From the Newton-Leibniz formula we see that positivity of the derivative and
imply
). The derivative
(1)
is positive, so is increasing.
Thus, is a distribution function, and it generates a density (1).
Example 3. Distribution function and density of a discrete variable
The distribution function concept applies to all random variables, both discrete and continuous. For discrete variables, the distribution function is not continuous as in Figure 1 here; it has jumps at points that have a positive probability attached. We illustrate this using a Bernoulli variable such that
and
.
- For
we have
.
- For
we have
.
- Finally,
for
.
This leads us to Figure 1.

Figure 1. Distribution function of the Bernoulli variable
Now consider such that
and
. The analog of the density function for Bernoulli looks like this:
(2) for
.
To understand this equation, check that and
. In Math, there are many tricks like this.
Remark. For continuous random variables, the value of the density at a fixed point means nothing (in particular, it can be larger than 1). It is its integral that has probabilistic meaning. For (2) the value of the density at a fixed point IS probability.
Leave a Reply
You must be logged in to post a comment.