The magic of the distribution function
Let be a random variable. The function where runs over real numbers, is called a distribution function of In statistics, many formulas are derived with the help of The motivation and properties are given here.
Oftentimes, working with the distribution function is an intermediate step to obtain a density using the link
A series of exercises below show just how useful the distribution function is.
Exercise 1. Let be a linear transformation of that is, where and Find the link between and Find the link between and
The solution is here.
The more general case of a nonlinear transformation can also be handled:
Exercise 2. Let where is a deterministic function. Suppose that is strictly monotone and differentiable. Then exists. Find the link between and Find the link between and
Solution. The result differs depending on whether is increasing or decreasing. Let's assume the latter, so that is equivalent to Also for simplicity suppose that for any Then
Differentiation of this equation produces
(the derivative of is negative).
For an example when is not invertible see the post about the chi-squared distribution.
Exercise 3. Suppose where and are independent, have densities and What are the distribution function and density of
Solution. By independence the joint density equals so
(converting a double integral to an iterated integral and remembering that is zero on the left half-axis)
Now by the Leibniz integral rule
A different method is indicated in Activity 4.11, p.207 of J.Abdey, Guide ST2133.
Exercise 4. Let be two independent random variables with densities . Find and
See this post.
Exercise 5. Let be two independent random variables. Find and
Solution. The inequality holds if and only if both and hold. This means that the event coincides with the event It follows by independence that
For we need one more trick, namely, pass to the complementary event by writing
Now we can use the fact that the event coincides with the event Hence, by independence
Equations (2) and (3) can be differentiated to obtain the links in terms of densities.