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I am trying to solve a particular problem involving multiple conditions on a random variable. We have $X_1, X_2....$ such that $X|K = k_i$ is a negative binomial with number of success given by $max(k-k_i,0)$ with success probability q where k is fixed. Random variable K is conditioned on Y and $Y$ is $ Bin(n,p) $ such that $(K|Y=y)$ is $Bin(Y, 1-\delta)$ From what I have read in the literature, K should be a Binomial Random Variable $Bin(n,pq)$

I have tried deriving the CDF of X using the law of total probability. But how should I account for that random number of success? Could somebody please guide me through the algebra or provide me with a relevant link.

Resting Platypus
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  • (1) Do you want the Negative Binomial to be the total number of trials or the number of failures before the specified number of success $\max(k-k_i, 0)$? (2) When $k_i \leq k$, how would you define the Negative Binomial distribution with the parameter $\max(k-k_i, 0)$ which is zero? Do you consider it a degenerate distribution that is always one (total number of trials) or always zero (number of failures)? – Lee David Chung Lin Dec 06 '18 at 10:44
  • I want the Negative Binomial to be the total number of trials. 2) About this question, I don't quiet understand your point. Could you please elaborate on that. Maybe there is something I haven't thought about.
    • – Resting Platypus Dec 06 '18 at 15:51
  • Sorry that was a typo on my part. I meant to ask that when $k \leq k_i$ (I mistakenly flipped the sign in the last comment) such that $\max(k-k_i, 0) = 0$. So you want the Negative Binomial to be the count when you have ... zero success? Then you would have to define it to be a degenerate 0 or 1. – Lee David Chung Lin Dec 07 '18 at 01:02
  • Yes. I want the NB to count the zero success as well. In my project X is actually the duration. So the duration can be zero if ki = k. – Resting Platypus Dec 07 '18 at 17:44