Table 2 Frequency distribution of the α- particles emitted by a radioactive substance in 2608 periods (Rutherford and Geiger, 1910)
From: Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution
x | Observed | Expected Frequencies | |||
|---|---|---|---|---|---|
HP (λ, β) | MLFD (λ, α, 1) | MLFD (λ, α, β) | COMNB (λ, α, β) | ||
0 | 57 | 49.916 | 48.766 | 56.731 | 40.352 |
1 | 203 | 206.489 | 202.884 | 198.608 | 202.736 |
2 | 383 | 408.368 | 406.600 | 393.766 | 420.265 |
3 | 525 | 530.655 | 533.254 | 527.594 | 543.646 |
4 | 532 | 513.471 | 518.076 | 523.971 | 515.429 |
5 | 408 | 395.776 | 398.948 | 408.328 | 389.504 |
6 | 273 | 253.493 | 254.119 | 259.549 | 246.804 |
7 | 139 | 138.885 | 137.888 | 138.392 | 135.597 |
8 | 45 | 66.481 | 65.122 | 63.227 | 66.139 |
9 | 27 | 28.253 | 27.212 | 25.167 | 29.143 |
10 | 10 | 10.796 | 10.192 | 8.846 | 11.756 |
11 | 4 | 3.748 | 3.457 | 2.776 | 4.387 |
12 | 0 | 1.192 | 1.071 | 0.785 | 1.527 |
13 | 1 | 0.350 | 0.306 | 0.202 | 0.499 |
14 | 1 | 0.127 | 0.106 | 0.059 | 0.217 |
Total | 2608 | 2608 | 2608 | 2608 | 2608 |
\( \widehat{\lambda} \) | 3.789(0.101) | 4.228(0.286) | 10.166 (7.06) | 5.284(16.70) | |
\( \widehat{\alpha} \) | ---- | 1.037(0.028) | 1.281 (0.182) | 0.951(3.121) | |
\( \widehat{\beta} \) | 0.916(0.096) | ---- | 2.170 (1.044) | 1.527(0.921) | |
χ 2, df | 18.382, 11 | 11.367, 10 | 8.179, 9 | 22.326, 9 | |
p-val | 0.073 | 0.330 | 0.516 | 0.008 | |
AIC | 10707.500 | 10706.412 | 10705.400 | 10718.180 | |