# Some Approximations to the Binomial Distribution Function

```@article{Bahadur1960SomeAT,
title={Some Approximations to the Binomial Distribution Function},
journal={Annals of Mathematical Statistics},
year={1960},
volume={31},
pages={43-54}
}```
• 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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