# 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.