Negative Binomial: Y ∼ N B ( r , p ) for r = number of success and p is the probability of each success p ( k ) = p ( Y = k ) = ( r − 1 k − 1 ) p r ( 1 − p ) k − r , r ≥ kthen P r ( l ≤ Y ≤ k ) = P r ( Y ≤ k ) − P ( Y ≤ l − 1 ) = P r ( N k ≥ r ) − P r ( N l − 1 ≥ r )where N k is the number of successes in the first k trials and N l − 1 the number of successes in the first l-1 trials.I'm not understanding how theses equate to each other, why is it greater than equal to? P r ( Y ≤ k ) − P ( Y ≤ l − 1 ) = P r ( N k ≥ r ) − P r ( N l − 1 ≥ r )

Question

Negative Binomial:   Y  ∼  N  B  (  r  ,  p  ) for   r  = number of success and p is the probability of each success  p  (  k  )  =  p  (  Y  =  k  )  =      (          r      −      1              k      −      1        )      p    r    (  1  −  p      )          k      −      r        ,  r  ≥  kthen  P  r  (  l  ≤  Y  ≤  k  )  =  P  r  (  Y  ≤  k  )  −  P  (  Y  ≤  l  −  1  )  =  P  r  (      N    k    ≥  r  )  −  P  r  (      N          l      −      1        ≥  r  )where       N    k   is the number of successes in the first k trials and       N          l      −      1       the number of successes in the first l-1 trials.I&#039;m not understanding how theses equate to each other, why is it greater than equal to?  P  r  (  Y  ≤  k  )  −  P  (  Y  ≤  l  −  1  )  =  P  r  (      N    k    ≥  r  )  −  P  r  (      N          l      −      1        ≥  r  )

Vicente Macias · Accepted Answer

Step 1Y counts the number of trials until r successes. Nk counts the successes in the first k trials.  Y  ≤  k is the event of r successes occurring in no more than k trials.      N    k    ≥  r is the event of r or more successes in the first k trials.Step 2These are the same event, that the r-th success does not occur after the k-th trial.So their probabilities had better be equal.  ∴        P    (  Y  ≤  k  )  −      P    (  Y  ≤  l  −  1  )  =      P    (      N    k    ≥  r  )  −      P    (      N          l      −      1        ≥  r  )

Negative Binomial: Y ~ NB (r,p) for r = number of success and p is the probability of each success p(k)=p(Y=k)=((k-1),(r-1)) p^r(1-p)^{k-r}, r >= k

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