yields 0.001079819330263761
The exact answer is:
yields 0.0010798643294
Seems pretty good. Try for larger n with x small in comparison to convince yourself numerically.
Have you tried getting the limit of the ratio of the two expressions (the exact one and the approximate one)? It does not approach 1.
I think the limit is 1.
According to M that is true. Why do you think the limit is not 1?
To prove that you might need the limit but maybe since Stirlings formula is asymptotic to the factorial it might be implied in step 3.
Stirlings is an asymptotic form for the factorial. The limit of the ratio of Stirlings and the factorial is 1. The fact that he use Stirlings in his proof guarantees the above limit.
]]>Have you tried getting the limit of the ratio of the two expressions (the exact one and the approximate one)? It does not approach 1.
]]>I am not getting much of the derivation either. It is a lot of algebra and undoubtedly was done with the help of a package. I put it down so you would have something.
]]>The paper I am looking at "Gaussian and Coins."
Using Stirlings:
Notice the approximately equal sign that is because you are approximated a discrete distribution ( binomial ) with the Normal distribution.
1) is an approximation for 2) which the above steps prove. Even for large N it is still an approximation. When N approaches infinity 1) = 2).
To prove that you might need the limit but maybe since Stirlings formula is asymptotic to the factorial it might be implied in step 3.
]]>Have you tried taking the limit as N goes to infinity of the ratio of the exact answer and the approximate one and proving it equals 1?
]]>Do what?
The answer is
that I can prove.
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