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I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF?


$^{(1)}$ For example, some are given in this paper.


To clarify my question taking advantage of the comments, I am looking for a polynomial of degree $n$, $P_n(x)$ such that, if $F(x)$ is the CDF of the standard Normal, then $F(x) \approx P_n(x)$ for $x$ in a suitable range, say $[-3,3]$.

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Why not differentiate the "several polynomial approximations to the Normal CDF" that you have found? – J. M. Aug 15 '12 at 12:46
@J.M. My mistake, you are absolutely right. I don't know why, but I implicitly assumed it would not have been a good approximation. – Libra Aug 15 '12 at 13:45
EDIT: Actually, just few of the approximations I have found are polynomial. So, I edited the original question. – Libra Aug 15 '12 at 14:53
@Libra Is this question now deemed answered? If so, write an answer and accept it. Maybe give an example, to make it more complete. – M Turgeon Aug 15 '12 at 14:54
And then comes the bad news: for every nonconstant polynomial $P$ and every CDF $F$, the function $(F-P)$ is unbounded. So it seems that, as I said, the assertion that $F(x)\approx P_n(x)$ is in serious need of some context. – Did Aug 15 '12 at 16:32

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