Some Approximations to the Binomial Distribution Function

  title={Some Approximations to the Binomial Distribution Function},
  author={R. R. Bahadur},
  journal={Annals of Mathematical Statistics},
  • R. R. Bahadur
  • Published 1 March 1960
  • Mathematics
  • Annals of Mathematical Statistics
Let p be given, 0 n (k) = ∑ n r=k ( n r )p r q n-r , where q = 1 - p. It is shown that B n (k) = [( n k ) p k ,q -k ] qF(n + 1, 1; k + 1; p), where F is the hypergeometric function. This representation seems useful for numerical and theoretical investigations of small tail probabilities. The representation yields, in particular, the result that, with A n (k) = [( n k )p k q n-k+I J [(k + 1)/(k + 1 - (n + l)p)], we have 1 ≤ A n (k)/B n (k) ≤ 1 + x -2 , where x = (k - np)/(npq) t . Next, let N n… 

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