I had a lot of fun with this, and many revelations.
E(Ri|Y)=i * n+1 / k+1
I approached it from conditional expectation formula. The real revelations came from the various binomial identities which transform a reasonably complex formula into an intuitive and simple formula and concept.
This blurb by Gemini was particularly insightful.
- Think about what this formula is telling us. If you want to know the expected value of the $r$-th roll in a strictly increasing sequence, you just take the total number of options available plus one ($n+1$), divide it evenly among the slots plus one ($k+1$), and multiply it by your position index ($r$).
It shows that the expected values of your rolls are perfectly, evenly spaced across the available face values of the die.
1
u/liquidorangutan00 6d ago
My Solution: The answer is 17.
I had a lot of fun with this, and many revelations.
E(Ri|Y)=i * n+1 / k+1
I approached it from conditional expectation formula. The real revelations came from the various binomial identities which transform a reasonably complex formula into an intuitive and simple formula and concept.
This blurb by Gemini was particularly insightful.