Question

Given a fair four-sided die, let Y equal the number of rolls needed to observe each face at least once.

Argue that Y = X₁ + X₂ + X₃ + X₄, where Xi has a geometric distribution with pi = (5−i)/4, i = 1, 2, 3, 4, and X₁, X₂, X₃, X₄ are independent.

Answer 1

The number Y equals the sum of variables X_1 to X_4, which represent the rolls needed to see each new face at least once on a four-sided die. Each X_i variable follows a geometric distribution, with decreasing probabilities as more unique faces are observed.

Step-by-step explanation:

The question relates to the concept of the geometric distribution within probability theory, which is a discrete probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials.


Each roll of the die is independent, and the probability of success changes after each successful roll, because the set of outcomes that count as a success becomes smaller as the faces of the die appear.

To see that Y = X1 + X2 + X3 + X4.

We define Xi as the number of rolls to get the i-th new face after having observed i-1 distinct faces.


For the first roll, any face will be new, so X1 follows a geometric distribution with p1 = 1 (since all faces are new one); this is equivalent to saying we are guaranteed to see a new face on the first roll.


After the first face has been observed, there are three new faces left, and the probability to see a new one is 3/4, hence X2 has a geometric distribution with p2 = 3/4.


Similarly, X3 has a geometric distribution with p3 = 2/4 = 1/2, and X4 with p4 = 1/4.