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
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
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.
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.