1. [Exercise 5.2.1 and Exercise 5.2.16 in Hogg/Tanis] It is not difficult to find the p.d.f. for the sum of the rolls of two fair 4-sided dice. It is more challenging to find the p.d.f. of the sum of the rolls of twelve 4-sided dice, but a convolution program can do this easily. Solution

2. [Exercise 5.2.4 and Exercise 5.2.17 of Hogg/Tanis] The power of the probability-generating function (factorial moment-generating function) for determining a p.d.f. can be illustrated with a very simple problem. Let X equal the number of times that an n -sided die must be rolled to observe each face at least once. The probability-generating function of X , as a function of t , is . A CAS can expand h = h ( t ) and list the respective probabilities, P ( X = x ), the coefficients of i n the expansion. Solution

3. [Exercise 5.2.13 and Exercise 11.6.18 of Hogg/Tanis] 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