conditional waiting – The number of trials depends on the number of successes of the previous period

  • In each period $ i $, $ X_i $ are drawn from the binomial ($ N_i $,$ alpha $)
  • The number of tests in period $ i + 1 $ (that is to say.$ N_ {i + 1} $) depends on the number of successes of the period $ i $
  • In particular, $ N_ {i + 1} = X_i $
  • Further, $ N_1 = n $

In this stochastic process of $ X_i $, I would like to prove the following two statements

  1. $ E big[frac{1+X_1+X_2+…+X_t}{1+N_1+…+N_t}big]> alpha $
  2. $ E big[frac{1+X_1+X_2+…+X_t}{1+N_1+…+N_t}big]> E big[frac{1+X_1+X_2+…+X_{t+1}}{1+N_1+…+N_{t+1}}big]$

Intuitively, the two statements above should be verified. But I have trouble proving this mathematically.