## 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, differentiable and 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

For an example when

**Exercise 3**. Suppose

**Solution**. The joint density

(converting a double integral to an iterated integral and remembering that

Now by the Leibniz integral rule

(1)

A different method is indicated in Activity 4.11, p.207 of J. Abdey, Guide ST2133.

**Exercise 4**. Let

See this post.

**Exercise 5**. Let

**Solution**. The inequality

For

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