Maximum Probability - Incomplete Gamma Law - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (/forum-10.html) +--- Forum: HP Prime Software Library (/forum-15.html) +--- Thread: Maximum Probability - Incomplete Gamma Law (/thread-13353.html) Maximum Probability - Incomplete Gamma Law - Eddie W. Shore - 07-27-2019 01:32 PM Blog Post: http://edspi31415.blogspot.com/2019/07/hp-prime-and-ti-84-plus-ce-maximum.html The program IGLMAX calculates four calculation points of the Incomplete Gamma Law: Three parameters of A, γ, and β of the IGL (Incomplete Gamma Law): g(x) = 1 / (β^γ * gamma(γ)) * x^(γ - 1) * e^(x/β) where: X = list of data points, where x_i ≥ 0 s = number of data points r = number of points where x_i = 0 (zero points) A = ln(mean(X)) - (Σ ln(X_i))/(s - r) γ = 1/(4*A) * (1 + √(1 + 4*A/3)) β = mean(X)/γ And p = probability that x is not exceeded p = r/s + (1 - r/s) * (1 - uigf(γ, x)/gamma(γ)) Gamma Function: gamma(γ) = ∫( t^(γ - 1) * e^(-t) dt, 0, ∞ ) Upper Incomplete Gamma Function: uigf(γ, x) = ∫( t^(γ - 1) * e^(-t) dt, x/β, ∞ ) One particular application is determining the maximum limit that rainfall exceeds x (inches or mm). The book "Pocket Computers in Agrometeorology" introduces this concept and provides a program for the classic TI-59 (see source below). HP Prime Program IGLMAX Code: ```EXPORT IGLMAX(L1,X) BEGIN // list of data ≥0, X // 2019-06-16 EWS LOCAL S,R,M,L,I; LOCAL A,Y,B,N,G,P; // set up S:=SIZE(L1); R:=0; // count zeros M:=0; // ΣX L:=0; // Σ(LN(X)) // counting loop FOR I FROM 1 TO S DO IF L1(I)==0 THEN R:=R+1; ELSE M:=M+L1(I); L:=L+LN(L1(I)); END; END; // parameters A:=LN(M/(S-R))-L/(S-R); Y:=(4*A)^(−1)*(1+√(1+4*A/3)); B:=M/(S-R)*1/Y; // gamma G:=CAS.Gamma(Y); // upper incomplete gamma N:=∫(T^(Y-1)*e^(−T),T,X/B,∞); // maximum probability P:=R/S+(1-R/S)*(1-N/G); // results RETURN {"A=",A,"γ=",Y, "β=",B,"Max Prob=",P}; END;``` Example: Data from a city of the rainfall in 2017 and 2018 (in inches): 2017 January: 3.90 February: 2.84 March: 2.31 April: 0.98 May: 0.64 June: 0.05 July: 0.00 August: 0.01 September: 0.00 October: 0.33 November: 0.72 December: 1.08 2018 January: 2.49 February: 2.66 March: 3.06 April: 2.94 May: 2.33 June: 0.81 July: 0.05 August: 0.00 September: 0.00 October: 0.14 November: 0.50 December: 2.24 Parameters: A: 0.7237035089 γ (Gamma): 0.8296776362 β (Beta): 1.812752248 Probability that X inches of rainfall will not exceed: X = 1 in: 0.593857 X = 2 in: 0.781173 X = 3 in: 0.879613 Source: R.A. Gommes "Pocket Computers In Agrometeorology" Food and Agriculture Organization of the United Nations. FAO PLANT PRODUCTION AND PROTECTION PAPER. Rome, 1983. ISBN 92-5-101336-5