A density function can give rise to a distribution function, and conversely, a distribution function can generate a density function. This is the thrust of this post, with an application to binary choice models to follow.
From distribution function to density function
Recall that for any random variable , its distribution function
is defined. As the next definition contains an existence requirement, in general
may not have a density (even when it is continuous).
Definition. We say that a random variable has a density
if there exists an integrable function
such that one can find the value of the distribution function by integrating
:
(1) for all real
.
When does have a density, most its values are where the density is the highest. The analog of this observation for the distribution functions looks as follows.

Figure 1. Relationship between density and distribution functions
Most values of
Consequences
Interval formula. Since the integral is additive, the interval formula in terms of the distribution function implies an interval formula in terms of the density function:
(2)
for any
Integral of density. Letting in (1)
Nonnegativity. Since the left side in (2) is nonnegative for any
Rule to find density. The Newton-Leibnitz formula says that the derivative of an integral with respect to the variable upper limit is the integrand evaluated at that limit:
Applying this to (1) we see that if the density exists, it can be found by differentiating the distribution function:
From density function to distribution function
Suppose we have a function
Remark. Since
Leave a Reply
You must be logged in to post a comment.