Question: How to extract coefficient from Probability Generating function

July 04 2012 dj_gssst 10
Maple

1

Dear experts,

 I want to extract the coefficient of z^k from an infinite series summation (known as probability generating function or characteristic function or Z-transformation) , the co-efficient is actually the probability of the R.V. will take value k.

Next problem in the same question is the summation is not working if I sum from 0 to infinity. It is working from 0 to 300.

I have attached the PDF for expressing the problem and the Maple code. I have used convert(.., FormalPowerSeries,.. );

Question_FPS.pdfQuestion_FPS.pdfQuest_FormalPowerSer.mw

restart; with(LinearAlgebra); with(RootFinding); Digits := 100; lambda := 3; mu := 7; eta := 4; xi := 2

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7

 

4

 

2

(1)

A := proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc;

proc (z) options operator, arrow; lambda^5/(lambda+mu-mu*z)^5 end proc

 

proc (z) options operator, arrow; lambda^5/(lambda+xi+eta-eta*z)^5 end proc

 

proc (z) options operator, arrow; xi*(A(z)-B(z))/(xi-(mu-eta)*(1-z)) end proc

(2)

expand(A(z), z);

243/(10-7*z)^5

 

Sum(((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(3)

convert(B(z), FormalPowerSeries, z);

Sum(((1/243)*(4/9)^k+(25/2916)*(4/9)^k*k+(35/5832)*(4/9)^k*k^2+(5/2916)*(4/9)^k*k^3+(1/5832)*(4/9)^k*k^4)*z^k, k = 0 .. infinity)

(4)

convert(C(z), FormalPowerSeries, z);

Sum((-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)*z^k, k = 0 .. infinity)

(5)

a := proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc; c := proc (k) options operator, arrow; -(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4 end proc

proc (k) options operator, arrow; (243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4 end proc

 

proc (k) options operator, arrow; -(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4 end proc

(6)

delta := .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182;

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786

 

.1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4)

 

.1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4)

 

proc (k) options operator, arrow; .1048169575235145793318440543701519642440665106256152261908658517046234684660798162819355335946881182^(k-1)*((243/100000)*(7/10)^k+(81/16000)*(7/10)^k*k+(567/160000)*(7/10)^k*k^2+(81/80000)*(7/10)^k*k^3+(81/800000)*(7/10)^k*k^4) end proc

 

proc (k) options operator, arrow; .1245100524719697889905669478100306655138374302028490892546160960154849333857498355256653408331912786^k*(-(15825128/1564031349)*(4/9)^k-(2312806/204004089)*(4/9)^k*k-(42167/8869743)*(4/9)^k*k^2-(338/385641)*(4/9)^k*k^3-(1/16767)*(4/9)^k*k^4+(4340868651/321817150000)*(7/10)^k+(153080361/11193640000)*(7/10)^k*k+(208089/38934400)*(7/10)^k*k^2+(3969/4232000)*(7/10)^k*k^3+(567/9200000)*(7/10)^k*k^4) end proc

(7)

sum(delta^(k-1)*a(k), k = 1 .. 20);

0.1075168698791322945083201474910417899915695562887365123785014410740837202600315477669688378727669084e-1

 

0.1075168698791322945083201474910417899915695562887365123785014410740837202600315477669688378727669084e-1

 

0.1075168698791322945131368855542654932195602557245309692494169359920231557707996781154291830071959580e-1

(8)

Warning,  computation interrupted

 

sum(theta^k*c(k), k = 1 .. 200); sum(g(k), k = 1 .. 200); sum(g(k), k = 1 .. 300)

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

 

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

 

0.1854156167422363765076984332241875155540949949705316457513708419061619470578009411403103805950641336e-2

(9)

Warning,  computation interrupted

 

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