Research Article Volume 5 Issue 4
Department of Mathematical Sciences, Tezpur University, India
Correspondence: Junali Hazarika, Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur, Assam, India
Received: January 26, 2017 | Published: April 7, 2017
Citation: Borah M, Hazarika J. Discrete shanker distribution and its derived distributions. Biom Biostat Int J. 2017;5(4):146-153. DOI: 10.15406/bbij.2017.05.00140
One parameter continuous Shanker distribution introduced by Shanker (2015 b) with parameter is defined by its probability density function (pdf).
f(x:θ)= θ2θ2+1 (θ+x) e−θx. x>0. θ>0.f(x:θ)= θ2θ2+1 (θ+x) e−θx. x>0. θ>0. (1.1)
Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which takes values in {0,1, . . .} {0,1, . . .} . This new distribution is generated by discretizing the continuous survival function of the Shanker distribution, which is may be obtained as
S(x)=∞∫xf(x:θ)dxS(x)=∞∫xf(x:θ)dx
= θ2+1+ θxθ2+1 e−θx, x>0. θ>0.= θ2+1+ θxθ2+1 e−θx, x>0. θ>0. (2.1)
S(x+1)= θ2+1+ θ(x+1)θ2+1 e−θ(x+1), x>0. θ>0.S(x+1)= θ2+1+ θ(x+1)θ2+1 e−θ(x+1), x>0. θ>0. (2.2)
The probability mass function (pmf) of discrete Shanker distribution may be obtained as
P(X=x)=S(x)−S(x+1)P(X=x)=S(x)−S(x+1)
=(θ2+1+ θx)(1− e−θ)−θ e−θθ2+1e(−θx),=(θ2+1+ θx)(1− e−θ)−θ e−θθ2+1e(−θx),
x=0, 1,2, 3x=0, 1,2, 3
(2.3)
Probability recurrence relation
Probability recurrence relation of discrete Shanker distribution may be obtained as
P(r+2)=e−θ(2Pr+1− e−θPr) , r ≥1P(r+2)=e−θ(2Pr+1− e−θPr) , r ≥1 (2.5)
Where P0=(θ2+1)(1− e−θ)−θ e−θθ2+1P0=(θ2+1)(1− e−θ)−θ e−θθ2+1 , and
P1=(θ2+1+ θ)(1− e−θ)−θ e−θθ2+1 e−θP1=(θ2+1+ θ)(1− e−θ)−θ e−θθ2+1 e−θ (2.6)
Factorial moment recurrence relation
Factorial moment generating function (fmgf) may be obtained as
M(t)=G(1+t)M(t)=G(1+t)
= (θ2+1)(1− e−θ)−θ(θ2+1)(1−e−θ− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θ− e−θt)2(θ2+1)(1− e−θ)−θ(θ2+1)(1−e−θ− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θ− e−θt)2 (2.7)
The more general form of factorial moment may also be written as
μ'[r]=r! e−θr[(θ2+1)(1−e−θ)+θr](θ2+1)(1−e−θ)r+1 (2.8)
If a random variable X have discrete Shanker distribution with parameter θ then the pmf of the size-biased distribution may be derived as
fs(x;θ)=xPxμ , x=1, 2, 3, … (3.1)
Where Px and μ denote respectively pmf and the mean of discrete Shanker distribution.
The pmf fs(x;θ) of size- biased discrete Shanker distribution with parameters θ may be derived from (3.1) as
fs(x,α)=xpxμ
=xe−θ(x−1)[(θ2+1+ θx)(1− e−θ)−θ e−θ](1−e−θ)2[(θ2+1)(1−e−θ)+θ] x=1,2,3,… (3.2)
Probability generating function Gs(t) for Size- biased Discrete Shanker Distribution may be obtained as
Gs(t)=∑∞x=0txPx
=t[(θ2+1)(1− e−θ)−θ e−θ](1− e−θ)2(1− e−θt)+θ(1− e−θ)3(1+e−θt)[(θ2+1)(1− e−θ)+θ](1− e−θt)3 (3.3)
Probability recurrence relation for size- biased discrete shanker distribution
Probability recurrence relation of Size- biased Discrete Shanker Distribution distribution may be obtained as
Pr=e−θ[3Pr−1−3e−θPr−2+e−θPr−3] for r>2 (3.4)
where
P1=[(θ2+1+ θ)(1− e−θ)−θ e−θ](1−e−θ)2[(θ2+1)(1−e−θ)+θ] and (3.5)
P2=2e−[(θ2+1+2θ)(1− e−θ)−θ e−θ](1−e−θ)2[(θ2+1)(1−e−θ)+θ]
Factorial moment recurrence relation for size- biased discrete shanker distribution
Factorial moment generating function Ms(t) of Size- biased discrete Shanker distribution may be obtained as
Ms(t)=Gs(1+t)
M(t)=(1+t)[(θ2+1)(1− e−θ)−θ e−θ](1− e−θ)2(1− e−θ−e−θt)+θ(1− e−θ)3(1+e−θ+e−θt)[(θ2+1)(1− e−θ)+θ](1−e−θ− e−θt)3 (3.6)
More general form μ[r]=r! e−θ(r−1)[(θ2+1)(1−e−θ)(r+e−θ)+θ(r2−e−θ)][(θ2+1)(1− e−θ)+θ](1−e−θ)r (3.7)
Factorial moment recurrence relation of Size- biased discrete Shanker distribution may be obtained as
μ'[r]=e−θA3[3A2rμ'[r−1]−3Ar(r−1) e−θ μ'[r−2]+Ar(r−1)(r−2)e−2θ μ'[r−3]], (3.7)
Where A = 1−e−θ.
μ[1]=[(θ2+1)(1+e−θ)+θ][(θ2+1)(1− e−θ)+θ]
μ[2]=2e−θ[(θ2+1)(1−e−θ)(2+e−θ)+θ(4−e−θ)][(θ2+1)(1− e−θ)+θ](1−e−θ)2 (3.8)
μ[3]=6 e−2θ[(θ2+1)(1−e−θ)(3+e−θ)+θ(9−e−θ)][(θ2+1)(1− e−θ)+θ](1−e−θ)3
The parameter θ of Shanker distribution has been estimated using Newton’s –Raphson method by considering appropriate initial guest value for θ . The function of θ can be expressed as
Shanker et al.1 fitted Poisson distribution (PD), Poisson- Lindley distribution (PLD) and Poisson-Akash distribution (PAD) to eleven numbers of data sets covering ecology, genetics and thunderstorms. In this investigation discrete Shanker (DS) distribution has been fitted to all 11 data sets have been considered for a comparison (Tables 1-11).2-36
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
213 |
213 |
202.1 |
234 |
236.8 |
1 |
128 |
109.15 |
138 |
99.4 |
95.6 |
2 |
37 |
47.44 |
47.1 |
40.5 |
39.9 |
3 |
18 |
19 |
10.7 |
16 |
16.6 |
4 |
3 |
7.25 |
1.8 |
6.2 |
6.7 |
5 |
1 |
2.67 |
0.2 |
2.4 |
2.7 |
6 |
0 |
1.48 |
0.1 |
1.5 |
1.7 |
Total |
400 |
400 |
400 |
400 |
400 |
|
Estimated θ |
1.1621 |
0.6825 |
1.950236 |
2.260342 |
χ2 |
7.89 | 10.08 |
11.04 |
14.68 |
|
d.f. |
3 | 2 |
2 |
2 |
|
p- vale |
0.0468 | 0.0065 |
0.004 |
0.0006 |
Table 1 Observed and expected number of Homocytometer yeast cell counts per square observed by Gosset
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
38 |
38 |
25.3 |
35.8 |
36.3 |
1 |
17 |
22.4 |
29.1 |
20.7 |
20.1 |
2 |
10 |
11.02 |
16.7 |
11.4 |
11.2 |
3 |
9 |
4.97 |
6.4 |
6 |
6.1 |
4 |
3 |
2.13 |
1.8 |
3.1 |
3.2 |
5 |
2 |
0.88 |
0.4 |
1.6 |
1.6 |
6 |
1 |
0.36 |
0.2 |
0.8 |
0.8 |
7 |
0 |
0.14 |
0.1 |
0.6 |
0.7 |
Total |
80 |
80 |
80 |
80 |
80 |
Estimated θ |
1.0494 |
1.15 |
1.255891 |
1.620588 |
|
χ2 |
6.246 |
18.27 |
2.47 |
2.07 |
|
d.f. |
4 |
2 |
3 |
3 |
|
p- vale |
0.1815 |
0.0001 |
0.4807 |
0.558 |
Table 2 Observed and expected number of red mites on Apple leaves
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
188 |
187.98 |
169.4 |
194 |
196.3 |
1 |
83 |
85.03 |
109.8 |
79.5 |
76.5 |
2 |
36 |
33.03 |
35.6 |
31.3 |
30.8 |
3 |
14 |
11.88 |
7.8 |
12 |
12.4 |
4 |
2 |
4.07 |
1.2 |
4.5 |
4.9 |
5 |
1 |
1.99 |
0.2 |
2.7 |
3.1 |
Total |
324 |
324 |
324 |
324 |
324 |
|
Estimated θ |
1.2644 |
0.648148 |
2.043252 |
2.345109 |
χ2 |
0.367 |
15.19 |
1.29 |
2.33 |
|
d.f. |
3 |
2 |
2 |
2 |
|
p- vale |
0.9470 |
0.0005 |
0.5247 |
0.3119 |
Table 3 Observed and expected number of European corn-border of McGuire et al
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
268 |
268 |
231.3 |
257 |
260.4 |
1 |
87 |
87 |
92.8 |
126.7 |
93.4 |
2 |
26 |
28.23 |
34.7 |
32.8 |
32.1 |
3 |
9 |
8.01 |
6.3 |
11.2 |
11.5 |
4 |
4 |
2.18 |
0.8 |
3.8 |
4.1 |
5 |
2 |
0.58 |
0.1 |
1.2 |
1.4 |
6 |
1 |
0.15 |
0.1 |
0.4 |
0.5 |
7 |
3 |
0.05 |
0.1 |
0.2 |
0.3 |
Total |
400 |
400 |
400 |
400 |
400 |
Estimated |
1.4870 | 0.5475 |
2.380442 |
2.659408 |
|
χ2 |
6.417 | 38.21 |
6.21 |
4.17 |
|
d.f. |
3 | 2 |
3 |
3 |
|
p- vale |
0.0930 | 0 |
0.1018 |
0.2437 |
Table 4 Distribution of number of Chromatid aberrations
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
413 |
413.01 |
374 |
405.7 |
409.5 |
1 |
124 |
134.97 |
177.4 |
133.6 |
128.7 |
2 |
42 |
38.92 |
42.1 |
42.6 |
42.1 |
3 |
15 |
10.49 |
6.6 |
13.3 |
13.9 |
4 |
5 |
2.71 |
0.8 |
4.1 |
4.6 |
5 |
0 |
0.68 |
0.1 |
1.2 |
1.5 |
6 |
2 |
0.22 |
0 |
0.5 |
0.7 |
Total |
601 |
601 |
601 |
601 |
601 |
|
Estimated |
1.5385 | 0.47421 |
2.685373 |
2.915059 |
χ2 |
5.562 | 48.17 |
1.34 |
0.29 |
|
d.f. |
3 | 2 |
3 |
3 |
|
p- vale |
0.1350 | 0 |
0.7196 |
0.9619 |
Table 5 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-60 μg/kg
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
200 |
200.01 |
172.5 |
191.8 |
194.1 |
1 |
57 |
70.01 |
95.4 |
70.3 |
67.6 |
2 |
30 |
21.51 |
26.4 |
24.9 |
24.5 |
3 |
7 |
6.17 |
4.9 |
8.6 |
8.9 |
4 |
4 |
1.69 |
0.7 |
2.9 |
3.2 |
5 |
0 |
0.45 |
0.1 |
1 |
1.1 |
6 |
2 |
0.16 |
0 |
0.5 |
0.6 |
Total |
300 |
300 |
300 |
300 |
300 |
|
Estimated |
1.4798 |
0.55333 |
2.35334 |
2.62674 |
χ2 |
8.191 |
29.68 |
3.91 |
3.12 |
|
d.f. |
3 |
2 |
2 |
2 |
|
p- vale |
0.0422 |
0 |
0.1415 |
0.2101 |
Table 6 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-75
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
155 |
155.01 |
127.8 |
158.3 |
160.7 |
1 |
83 |
82.63 |
109 |
77.2 |
74.3 |
2 |
33 |
37.2 |
46.5 |
35.9 |
35.3 |
3 |
14 |
15.41 |
13.2 |
16.1 |
16.5 |
4 |
11 |
6.08 |
2.8 |
7.1 |
7.5 |
5 |
3 |
2.32 |
0.5 |
3.1 |
3.3 |
6 |
1 |
1.35 |
0.2 |
2.3 |
2.4 |
Total |
300 |
300 |
300 |
300 |
300 |
|
Estimated |
1.1301 | 0.853333 |
1.617611 |
1.963313 |
χ2 |
3.432 | 24.97 |
1.51 |
1.98 |
|
d.f. |
4 | 2 |
3 |
3 |
|
p- vale |
0.4883 | 0 |
0.6799 |
0.5766 |
Table 7 Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-90
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
187 |
187.01 |
155.6 |
185.3 |
187.9 |
1 |
77 |
87.72 |
117 |
83.5 |
80.2 |
2 |
40 |
35.21 |
43.9 |
35.9 |
35.3 |
3 |
17 |
13.07 |
11 |
15 |
15.4 |
4 |
6 |
4.62 |
2.1 |
6.1 |
6.6 |
5 |
2 |
1.58 |
0.3 |
2.5 |
2.7 |
6 |
1 |
0.79 |
0.1 |
1.7 |
1.9 |
Total |
330 |
330 |
330 |
330 |
330 |
|
Estimated |
1.2345 | 0.751515 |
1.804268 |
2.139736 |
χ2 |
3.721 | 31.93 |
1.43 |
1.35 |
|
d.f. |
4 | 2 |
3 |
3 |
Table 8 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of June, January 1957 to December 1967
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
177 |
177.01 |
142.3 |
177.7 |
180 |
1 |
80 |
93.79 |
124.4 |
88 |
84.7 |
2 |
47 |
42.01 |
54.3 |
41.5 |
40.9 |
3 |
26 |
17.32 |
15.8 |
18.9 |
19.4 |
4 |
9 |
7.79 |
3.5 |
8.4 |
8.9 |
5 |
2 |
3.08 |
0.7 |
6.5 |
7.1 |
Total |
341 |
341 |
341 |
341 |
341 |
Estimated |
1.1348 | 0.8739 |
1.583536 |
1.938989 |
|
χ2 |
6.972 | 39.74 |
5.15 |
5.02 |
|
d.f. |
4 | 2 |
3 |
3 |
|
p- vale |
0.1374 | 0 |
0.1611 |
0.1703 |
Table 9 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of July, January 1957 to December 1967
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
185 |
184.99 |
151.8 |
184.8 |
187.5 |
1 |
89 |
92.42 |
122.9 |
87.2 |
83.9 |
2 |
30 |
39.27 |
49.7 |
39.3 |
38.6 |
3 |
24 |
15.39 |
13.4 |
17.1 |
17.5 |
4 |
10 |
6.74 |
2.7 |
7.3 |
7.6 |
5 |
3 |
2.19 |
0.5 |
5.3 |
5.9 |
Total |
341 |
341 |
341 |
341 |
341 |
|
Estimated |
1.1828 | 0.809384 |
1.693425 |
2.038417 |
χ2 |
8.987 | 49.49 |
5.03 |
4.69 |
|
d.f. |
4 | 2 |
3 |
3 |
|
p- vale |
0.0414 | 0 |
0.1696 |
0.196 |
Table 10 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for the month of August, January 1957 to December 1967
No. of yeast cell per square |
Observed |
Expected frequency |
|||
frequency |
DS |
PD |
PLD |
PAD |
|
0 |
549 |
549.01 |
449 |
547.5 |
555.1 |
1 |
246 |
274.27 |
364.8 |
259 |
249.2 |
2 |
117 |
117 |
116.54 |
148.2 |
116.9 |
3 |
67 |
45.67 |
40.1 |
51.2 |
52.3 |
4 |
25 |
17.04 |
8.1 |
21.9 |
23.2 |
5 |
7 |
7.16 |
1.3 |
9.2 |
10 |
6 |
1 |
2.31 |
0.5 |
6.3 |
7.3 |
Total |
1012 |
1012 |
1012 |
1012 |
1012 |
|
Estimated |
1.1828 | 0.812253 |
1.68899 |
2.033715 |
χ2 |
16.824 | 119.45 |
9.6 |
9.4 |
|
d.f. |
5 | 3 |
4 |
4 |
|
p- vale |
0.0048 | 0 |
0.0477 |
Table 11 Observed abd expected number of days that experienced X thunderstroms event at Cape Kennedy, Florida for 11-year period of record for Summer, January 1957 to December 1967
In this article, the discrete Shanker distribution has been introduced by discretizing the continuous Shanker distribution. We have studied some properties of the distributions. Further the applications of the distribution and goodness of fit of the distribution.
None.
None.
©2017 Borah, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.
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