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Suppose that a fair die is rolled repeatedly; different rolls are independent. N is the minimum number of rolls till each of the six faces of the die shows up at least once.

Suppose that i distinct faces have already shown up, i = 1; 2; 3; 4; 5. N_i is the minimum number of additional rolls after i distinct faces have shown up till another distinct face (one of the remaining 6 - i faces) shows up. Derive the pmf.

I know that N is Geometrically distributed by 1/6, however, I am having trouble figuring out if N_i is geometrically distributed like N, and why is that different if not? How would the p.m.f of N_i be derived?

zinde
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  • The answer to your question is apparent in the answer to the linked question. 2. No, your N is not geometrically distributed.
    • – Did Mar 16 '18 at 11:05
  • @Did: I cannot find the question referred to. – zoli Mar 16 '18 at 11:43
  • Quote: "After that, the random time until a third (different) result appears is geometrically distributed with parameter of success 4/6, hence with mean 6/4" Surely you can generalize this, no? – Did Mar 16 '18 at 11:46
  • @Did Thanks! I saw this page, but couldn't see the link to my question before. However, I think sourav's answer is closer to the answer to my question. – zinde Mar 16 '18 at 12:02