These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for Gamma Distribution bounded between [0,1].
pGAMMA(p,c,l)
p | vector of probabilities. |
---|---|
c | single value for shape parameter c. |
l | single value for shape parameter l. |
The output of pGAMMA
gives the cumulative density values in vector form.
The probability density function and cumulative density function of a unit bounded Gamma distribution with random variable P are given by
$$g_{P}(p) = \frac{ c^l p^{c-1}}{\gamma(l)} [ln(1/p)]^{l-1} $$ ; \(0 \le p \le 1\) $$G_{P}(p) = \frac{ Ig(l,cln(1/p))}{\gamma(l)} $$ ; \(0 \le p \le 1\) $$l,c > 0$$
The mean the variance are denoted by $$E[P] = (\frac{c}{c+1})^l $$ $$var[P] = (\frac{c}{c+2})^l - (\frac{c}{c+1})^{2l} $$
The moments about zero is denoted as $$E[P^r]=(\frac{c}{c+r})^l $$ \(r = 1,2,3,...\)
Defined as \(\gamma(l) \) is the gamma function. Defined as \(Ig(l,cln(1/p))= \int_0^{cln(1/p)} t^{l-1} e^{-t}dt \) is the Lower incomplete gamma function.
NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.
Olshen, A. C. Transformations of the Pearson Type III Distribution. Ann. Math. Statist. 9 (1938), no. 3, 176--200.
#plotting the random variables and probability values col <- rainbow(4) a <- c(1,2,5,10) plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values", xlim = c(0,1),ylim = c(0,4))#> [1] NaN 2.696915e-03 1.908823e-02 5.591507e-02 1.151860e-01 #> [6] 1.963513e-01 2.974469e-01 4.157526e-01 5.481913e-01 6.915796e-01 #> [11] 8.427858e-01 9.988302e-01 1.156946e+00 1.314613e+00 1.469580e+00 #> [16] 1.619861e+00 1.763742e+00 1.899766e+00 2.026722e+00 2.143630e+00 #> [21] 2.249727e+00 2.344447e+00 2.427404e+00 2.498378e+00 2.557294e+00 #> [26] 2.604211e+00 2.639301e+00 2.662839e+00 2.675188e+00 2.676786e+00 #> [31] 2.668131e+00 2.649777e+00 2.622316e+00 2.586375e+00 2.542603e+00 #> [36] 2.491664e+00 2.434233e+00 2.370984e+00 2.302590e+00 2.229714e+00 #> [41] 2.153006e+00 2.073098e+00 1.990604e+00 1.906109e+00 1.820178e+00 #> [46] 1.733342e+00 1.646106e+00 1.558942e+00 1.472288e+00 1.386552e+00 #> [51] 1.302105e+00 1.219288e+00 1.138405e+00 1.059728e+00 9.834979e-01 #> [56] 9.099206e-01 8.391724e-01 7.713992e-01 7.067177e-01 6.452166e-01 #> [61] 5.869582e-01 5.319796e-01 4.802945e-01 4.318943e-01 3.867500e-01 #> [66] 3.448137e-01 3.060204e-01 2.702894e-01 2.375261e-01 2.076239e-01 #> [71] 1.804654e-01 1.559242e-01 1.338667e-01 1.141532e-01 9.663983e-02 #> [76] 8.117977e-02 6.762468e-02 5.582605e-02 4.563641e-02 3.691057e-02 #> [81] 2.950669e-02 2.328725e-02 1.812007e-02 1.387906e-02 1.044499e-02 #> [86] 7.706093e-03 5.558644e-03 3.907329e-03 2.665574e-03 1.755734e-03 #> [91] 1.109170e-03 6.662063e-04 3.759598e-04 1.960545e-04 9.220202e-05 #> [96] 3.765617e-05 1.253703e-05 3.022169e-06 4.041950e-07 1.282574e-08 #> [101] 0.000000e+00#> [1] 0.334898#> [1] 0.02065365#plotting the random variables and cumulative probability values col <- rainbow(4) a <- c(1,2,5,10) plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values", xlim = c(0,1),ylim = c(0,1))#> [1] 1.000000e+00 9.999932e-01 9.998995e-01 9.995428e-01 9.987061e-01 #> [6] 9.971660e-01 9.947125e-01 9.911596e-01 9.863504e-01 9.801593e-01 #> [11] 9.724927e-01 9.632875e-01 9.525092e-01 9.401501e-01 9.262260e-01 #> [16] 9.107741e-01 8.938501e-01 8.755255e-01 8.558851e-01 8.350246e-01 #> [21] 8.130485e-01 7.900680e-01 7.661988e-01 7.415599e-01 7.162715e-01 #> [26] 6.904540e-01 6.642267e-01 6.377065e-01 6.110072e-01 5.842386e-01 #> [31] 5.575057e-01 5.309083e-01 5.045405e-01 4.784902e-01 4.528391e-01 #> [36] 4.276621e-01 4.030274e-01 3.789968e-01 3.556249e-01 3.329599e-01 #> [41] 3.110434e-01 2.899105e-01 2.695901e-01 2.501051e-01 2.314727e-01 #> [46] 2.137045e-01 1.968071e-01 1.807821e-01 1.656266e-01 1.513333e-01 #> [51] 1.378913e-01 1.252858e-01 1.134990e-01 1.025103e-01 9.229632e-02 #> [56] 8.283152e-02 7.408848e-02 6.603815e-02 5.865018e-02 5.189319e-02 #> [61] 4.573504e-02 4.014309e-02 3.508447e-02 3.052625e-02 2.643573e-02 #> [66] 2.278056e-02 1.952897e-02 1.664994e-02 1.411329e-02 1.188988e-02 #> [71] 9.951666e-03 8.271845e-03 6.824902e-03 5.586697e-03 4.534504e-03 #> [76] 3.647056e-03 2.904559e-03 2.288707e-03 1.782675e-03 1.371100e-03 #> [81] 1.040057e-03 7.770188e-04 5.708055e-04 4.115309e-04 2.905353e-04 #> [86] 2.003148e-04 1.344429e-04 8.748908e-05 5.493214e-05 3.307222e-05 #> [91] 1.894087e-05 1.021113e-05 5.108634e-06 2.325028e-06 9.348735e-07 #> [96] 3.173967e-07 8.433588e-08 1.521193e-08 1.353245e-09 2.142264e-11 #> [101] 0.000000e+00#> [1] 0.334898#> [1] 0.02065365#only the integer value of moments is taken here because moments cannot be decimal mazGAMMA(1.9,5.5,6)#> [1] 0.3670253