<p>Gamma and Int</p>

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GAMMA PDF & CDF Calculations

3 appVersion 0.99.7251

AuxGAMMA 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

Gamma function. Valid for Re(ζ)>0 ζ Γ z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : zp zg 0.5 zh *^: 1 κ Clear 16 mat 2 π * sqrt e zg -^* zp 2^* AuxGAMMA 1 el AuxGAMMA κ 1 + el z κ + / κ 114 sum + (* eval 21 line :

Adaptive Integral. Integrates f(x) for x=a..b with default TOL and MaxIters 1 f a b IntAdapt x Clear x f a b TOL MaxIters IntAdapt 21 line :

1 f a b T I IntAdapt S 0 : k 1 : N 1 : x a 11 mat : h b a - 2 / 11 mat : 15 mat fa fc fb 13 mat a f 11 mat a h 1 el + f 11 mat b f 11 mat 13 mat eval : s h 1 el fa 1 el 4 fc 1 el * + fb 1 el + 3 / * 11 mat eval : t T s 1 el UnitsOf * : 12 mat k 0 > (N I < (& φ x k el h k el 2 / + f x k el 3 h k el * 2 / + f 12 mat eval : σ h k el 6 / fa k el 4 φ 1 el * + fc k el + fc k el 4 φ 2 el * + fb k el + 12 mat * eval : vx fao fxo fbo vh 15 mat x k el fa k el fc k el fb k el h k el 15 mat eval : k k 1 - : σ 1 el σ 2 el + s k 1 + el - abs T < S S σ 1 el σ 2 el + + eval : k k 1 + : x k el x k el vh + : fa k el fxo : fc k el φ 2 el : fb k el fbo : h k el 0.5 vh * : s k el σ 2 el : 71 line k k 1 + : x k el vx : fa k el fao : fc k el φ 1 el : fb k el fxo : h k el h k 1 - el : s k el σ 1 el : 71 line N N 1 + : 13 mat if 51 line while S eval N I < RET.MaxIt 51 line :

Auxiliary RET function. x q RET.MaxIt q x MaxIters Rached. error if 11 line :

TOL 105 -^:

MaxIters 200 :

X 30.4 278.7 63.9 387.1 330 187.3 495.9 18 163.3 128.8 101 mat :

LNX X ln vectorize : 3.4144 5.6301 4.1573 5.9587 5.7991 5.2327 6.2064 2.8904 5.0956 4.8583 101 mat

X.mean X Mean : 208.34

LNX.mean LNX Mean : 4.9243

A X.mean ln LNX.mean - : 0.4149

α 14 A * / 114 A 3 / * + sqrt + (* : 1.3536

β X Mean α / : 153.918

PDF x g 1 β α^α Γ * / x α 1 -^e x β / -^* * :

CFD x_val G xx g 0 x_val 105 -^200 IntAdapt 11 line :

CFD x_val G x g x 0 x_val int :

X 30.4 278.7 63.9 387.1 330 187.3 495.9 18 163.3 128.8 101 mat

X g vectorize 0.0034 0.0015 0.0035 0.0008 0.0011 0.0023 0.0004 0.003 0.0026 0.003 101 mat

X G vectorize 0.0825 0.7385 0.2 0.8595 0.8046 0.5676 0.9259 0.0425 0.5089 0.4133 101 mat

X G vectorize sum 5.1432890165

3 appVersion 0.99.7251