511 Reputation

18 years, 333 days

on infinite repeated function compositio...

Maple 13

Let

G:=s->q(1-ps)^(-1)

be a generating function with 1-p=q in (0,1). Then the repeated function composition

G(G(G(s)));(G@@3)(s);simplify(%);

gives me some concrete expression. What is the simplest way to get the general expression for 3 replaced by n?

Borel-Tanner moments...

Maple

For m=1,2, how do I show with Maple that the first two moments of the Borel-Tanner distribution are simple functions of k and lamda, e.g., k/(1-lambda) for the mean? How do I get the closed-form expressions with maple? Code:

simplify(sum(x^m*k*x^(x-k-1)*lambda^(x-k)*exp(-lambda*x)/factorial(x-k), x = k .. infinity)) assuming lambda > 0, lambda < 1, k::posint; evalf(subs(k = 1, lambda = .8, %))

DeMoivre's formula...

Maple

Consider the identity 4*cos(theta)^3-3*cos(theta) = cos(3*theta). Does maple know about it, or how to prove the identity, which is true by deMoivre's formula, with maple? The trig option for simplify seems not enough. Thanks.

solving a general cubic trig-ly...

Maple

There is some advice how to solve a cubic trigonometrically  here, but nothing presentable seems to appear out of it. Could someone please demonstrate how to do that with maple for a general (monic) cubic. Thanks.

Where is Maple mistaken?...

Maple

Why does the equality fail to hold in case of a positive geometric density (with the location parameter v=1,2,... if in general)? What's wrong: symbolical result or numerical evaluation? Thanks.
with(Statistics):
X := RandomVariable(Geometric(p))+1; M := Mean(X); A := simplify((X-M)/M, size);
Mean(1-A+A^2/(1+A))/M = (Mean(1)-Mean(A)+Mean(A^2/(1+A)))/M; is(%); subs(p = 1/3, `%%`); simplify(%, size); evalf(%)

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