Appendix 2: Proof of DeMoivre–Laplace Theorem
To prove the DeMoivre–Laplace theorem, a preliminary result, called in math language a “lemma,” makes things much easier.
Lemma A2.1
Suppose X is a random variable with moment-generating function MX(t). Further, suppose that, for fixed constants a and b, a ≠ 0, Y is another random variable such that Y = aX + b. Then
Proof
Assume X is discrete. Then Thus
Now, in the argument of the exponential, write k = a[(k − b)/a + b], to obtain
Now, use to rewrite the expression as
Now, write j = (k − b)/a and note that the sum must now be over all j, to ...
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