I was reading about sums of dice rolls and Chernoff bounds, and I thought of a question I couldn't obviously answer with the techniques I know.
You're given some number $x$ and told it was generated by summing subsequent (I.I.D.) rolls of an $n$-sided die, where the sides are numbered $1, 2, ..., n$. Let $X$ be a random variable representing the number of rolls that were summed.
What do we know about the distribution of $X$? Can we analyze it using Chernoff bounds?
To make it more concrete, say you're given the number 36 and told it is the sum of rolls of a standard six-sided die. It's very unlikely that it took 36 rolls for the sum to be 36 (the probability of 36 '1's in a row) and it's similarly unlikely that it only took six rolls (six '6's in a row - interestingly this is much more likely than the other extreme).
Thanks very much for your help.