I am trying to obtain the moments of this function via Probability Generating Function (PGF) or Moment Generating Function (MGF) but having problem with it. The pdf is
((Binomial[n, x])^v*\[Theta]^x*(1 - \[Theta])^(n - x))/\!\(\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(\*SuperscriptBox[\((Binomial[n, i])\), \(v\)]*
\*SuperscriptBox[\(\[Theta]\), \(i\)]*\*SuperscriptBox[\((1 - \[Theta])\), \(n - i\)]\)\)
I want to find the derivative of the PGF, with respect to t as expressed below and set t=1: that will give first moment. differentiating the second time and set t=1 gives second moment .......
D[\!\(\*UnderoverscriptBox[\(\[Sum]\), \(x = 1\), \(n\)]\*FractionBox[\(\*SuperscriptBox[\(t\), \(x\)]\ \*SuperscriptBox[\((1 - \[Theta])\), \(n - x\)]\ \*SuperscriptBox[\(\[Theta]\), \(x\)]\ \*SuperscriptBox[\(Binomial[n, x]\), \(v\)]\), \(\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\*SuperscriptBox[\((1 - \[Theta])\), \(\(-i\) + n\)]\ \*SuperscriptBox[\(\[Theta]\), \(i\)]\ \*SuperscriptBox[\(Binomial[n, i]\), \(v\)]\)]\), t]
I document is attached for details if this code is not clear enough. Thanks for your kindness.
Olorire
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