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Imagine that I have a coin that changes monotonically over time.

-- casino example --
(This is not necessary to understand mathematical problem, but just can help to imagine a real life situation, you can skip it)

Imagine that you are in a very clever casino The coin is made of insulator, one side is charged with positive charges and another with negative ones. Croupier flips the coin on a table that is made of a metal board covered with an insulator. Casino charged the metal board with positive charges. The croupier tosses the coin - it has higher probability to end 'positive up' (the negative charges on coin are attracted by positive ones on table) and I am winning. Then between tosses casino slightly charges the metal board with negative charges, so the coin becomes more likely to end 'negative up'. And with time I start loosing - good fortune seems gone.
-- end of example --

Let's note $p(t)$ the probability of tossing a tail in a discrete moment $t \ge 0$. We assume that:

  • $p(t+1) > p(t)$,
  • $\lim_{t\rightarrow\infty}p(t) = 1$

Questions:

A. How to calculate an expected number of coin tosses to get $n$ consecutive tails? This is a general question, we are just assuming $p(t)$ as above.

B. What would be a formula for this particular form of $p(t)$:

$$ p(t) = \frac{1}{1+e^{-\frac{1}{\tau}(t-t_0)}} $$ with $t_0 \ge 0$, $\tau > 0$. I am interested in a formula of a form: $$ N(n, t_0, \tau) = \dots $$

I am looking for answers to both questions. But if you only give me an answer to B., I will be very grateful too.

PS
This is a generalization of this question.

hans
  • 151
  • Welcome o MSE. Thanks for using MathJax. You tag this as "contest-math" please tell us what contest this comes from and supply a link to the contest, Also rather than having us reinvent the wheel, can you show any partial results you have? – Stephen Meskin Dec 05 '17 at 21:20
  • Tag was wrong. Sorry for confusion. I deleted it. Actually I need this result for predicting length of a learning process in deep learning. – hans Dec 05 '17 at 21:24
  • Thanks for clearing that up. But the last question still holds. Were you able to solve this in any special cases or is there any literature for special cases. Another issue that comes up is when a result seems to be needed to prove X but we don't know what X is. Sometimes X can be solved another way and the question posed turns out to be irrelevant and possibly unrealistic. So some more mathematical background would be useful. – Stephen Meskin Dec 05 '17 at 21:40
  • @StephenMeskin Thanks for your pertinent comment. 1. An obvious special case is the question I linked. 2. I couldn't find any literature about this specific problem - I was hitting variants of simpler problem. If someone can, I will appreciate it. 3. I suppose that X can be solved in another way, but the problem I posted seems interesting for me from mathematical point of view, even when finally it will not find application in my research. – hans Dec 05 '17 at 21:58
  • Let me keep pressing. Why the special form of $p(t)$ in part 2? Have you solved it for any other special forms of $p(t)$? The special case you mentioned previously does not satisfy the limit hypothesis of your question. – Stephen Meskin Dec 06 '17 at 19:00
  • @StephenMeskin that is the logistic function, and it pops up all over science. It does satisfy the monotonicity and limit conditions. – Larry B. Dec 08 '17 at 17:01
  • @LarryB. I was referring to https://math.stackexchange.com/a/364046/465208 which has a constant probability. – Stephen Meskin Dec 08 '17 at 18:12

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