Given a fair four-sided die, let Y equal the number of

Problem 15E

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

(a) Argue that  where  has a geometric distribution with  = (5 - i)/4, i = 1, 2, 3, 4, and  are independent.

(b) Find the mean and variance of Y.

(c) Find P(Y = y), y = 4, 5, 6, 7.

                                                        Step by Step Solution

Step 1 of 3

(a)

The number of independent trials required to obtain first success is geometric distribution.

Let  is the number of rolls needed to obtain 1 unique face,

Let  has the number needed to obtain the first face that is a different face obtained on the first roll.

Let  represents the number rolls needed to obtain the first face that is different from the previous two unique first roll,

Let  is different from the previous three unique faces.

So,  and .