A tetrahedron is a regular four-sided figure, like a four-sided die. Assume that when I roll a tetrahedron, each of the four sides is equally likely to appear, independent of all other rolls.

Suppose the sides of the tetrahedron are labeled 1, 2, 3, and 4. Define two random variables:

• is the label that I see on the first roll
• is the number of times I roll till I see a label I have seen before

For example, if the first few labels are 3, 4, 1, 3, 1, 2, 1 then X=3 and D=4.

a) What is the distribution of X?

b) What are the possible values of D?

c) Find the distribution of D.

d) Find the decimal values of the probabilities in Part c. What should their sum be?

e) What is the most likely value of D, and what is its probability?

f) Find (=2,=3)P(X=2,D=3).