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.

Now consider **density function for Bernoulli** looks like this:

(2)

To understand this equation, check that

**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.

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