Sums of Continuous Random Variables

Let , , ..., be a random sample of size n from a continuous distribution with p.d.f. f(x) , where f(x) > 0 for a < x < b and < a < x < b < . This section illustrates how MAPLE can be used to find the p.d.f. of the sum of these random variables. Example

1. Let f(x) = 1, 0 < x < 1. Show how to find the p.d.f. of . Solution

2. Let f(x) = 1, 0 < x < 1. Find the p.d.f. of + . . . + . Solution

3. Let f(x) = , < x < 1. Show how to find the p.d.f. of , , . Solution

4. Let f(x) = , < < 1. Find the p.d.f. of + . . . + .

Empirical Evidence
Theoretical Solution

5. A CAS can be used to show that the weighted average of two Cauchy random variables is Cauchy. This in turn can be used to prove that the distribution of X - bar is Cauchy when sampling from a Cauchy distribution. Solution