Biswas Distribution. Deapon Biswas. Transport Officer, Private Concern, Chittagong, Bangladesh

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American Journal of Mathematics and Statistics 2018, 8(4): 89-95 DOI: 10.592/j.ajms.20180804.02 Biswas Distribution Deapon Biswas Transport Officer, Private Concern, Chittagong, Bangladesh Abstract Here we get all the theoretical discrete distributions to the random experiment M sided N dice taken V at a time. It states joint probability functions as well as marginal probability functions of which number of possible outcomes is EB numbers and number of favorable outcomes is GB numbers. In this paper I introduce 20 definitions where first is the Biswas distribution and others are the related distributions that state different joint probability functions and marginal probability functions. Keywords Biswas distribution, SB distribution, AB distribution, SAB distribution, ASB distribution, First way B distribution, Second way B distribution, Both way B distribution, First way SB distribution, Second way SB distribution, Both way SB distribution, First way AB distribution, Second way AB distribution, Both way AB distribution, First way SAB distribution, Second way SAB distribution, Both way SAB distribution, First way ASB distribution, Second way ASB distribution, Both way ASB distribution 1. Introduction First of all Biswas distributions are divided into four parts i.e., (i) both way selected B distribution (SB distribution) (ii) both way arranged B distribution (AB distribution) (iii) first way selected and second way arranged B distribution (SAB distribution) (iv) first way arranged and second way selected B distribution (ASB distribution) And then each part divided into three parts again i.e., for B distribution we get (i) first way B distribution (ii) second way B distribution (iii) both way B distribution. 2. Findings Definition.1 Biswas distribution: A two dimensional random variable (, ) is said to follow Biswas distribution if it assumes only non negative values and its joint probability P(, ) b(, ; M, N, U, V) GBM N U EB M V N ; min(m, U) M+1 min(m, U, V) * Corresponding author: philosclub@yahoo.com (Deapon Biswas) Published online at http://journal.sapub.org/ajms Copyright 2018 The Author(s). Published by Scientific & Academic Publishing This work is licensed under the Creative Commons Attribution International License (CC B). http://creativecommons.org/licenses/by/4.0/ min(n, U) N+V min(n, U, V) M U and N U for first way and second way respectively. 0; otherwise. (.1) The four independent finite constants M, N, U and V are known as the parameters of this distribution. Biswas distribution is a joint probability distribution and a theoretical discrete distribution. Here takes only non-negative values under the interval min(m, U) M +1 min(m, U, V) and takes only non-negative values under the interval min(n, U) N+V min(n, U, V). Any two dimensional random variable which follows B distribution is known as B-variate and denoted by the symbol (, ) B(M, N, U, V). b(, ; M, N, U, V) GBM N U EB M V N EB M N GB M V V (.2) Example.1: For the GB experiment GB 5 6 find 4 2 1 P( 2, ) where the experiment is first way arranged and second way selected. Solution: We know P( 2, ) b(2, 1; 5, 6,, 4) GB5 6 4 2 1 EB 5 6 4 990 975 0.105. Definition.2 SB distribution: A two dimensional random variable (, ) is said to follow SB distribution if it assumes only non-negative values and its joint probability

90 Deapon Biswas: Biswas Distribution P(, ) sb(, ; M, N, U, V) GSBM V ESB M N V ;,, U and V value as (.1) 0; otherwise. (.) sb(, ; M, N, U, V) GSBM N U ESB M V N ESB M N GSB M V V (.4) Example.2: For the GSB experiment GSB 4 5 find (i) P( 2, ) and (ii) P(,1). Solution: We get (i) P( 2, ) sb(2, ; 4, 5,, ) GSB4 2 5 ESB 4 5 0.045. (ii) P(, ) sb(, 1; 4, 5,, ) GSB4 5 1 ESB 4 5 0.015. 9 200 200 Definition. AB distribution: A two dimensional random variable (, ) is said to follow AB distribution if it assumes only non-negative values and its joint probability P(, ) ab(, ; M, N, U, V) GABM N U EAB M V N ;,, U and V value as (.1) 0; otherwise. (.5) ab(, ; M, N, U, V) GABM N U EAB M V N EAB M N GAB M V V (.6) Example.: For the GAB experiment GAB 5 6 find 4 (i) P(1,) and (ii) P(2,2). Solution: We get (i) P(, ) ab(1, ; 5, 6,, 4) GAB5 4 6 1 14040 225000 0.0624. EAB 5 4 6 (ii) P( 2, 2) ab(2, 2; 5, 6,, 4) GAB5 6 4 2 2 EAB 5 4 6 71280 225000 0.16. Definition.4 SAB distribution: A two dimensional random variable (, ) is said to follow SAB distribution if it assumes only non-negative values and its joint probability P(, ) sab(, ; M, N, U, V) GSABM V ESAB M N V ;,, U and V value as (.1) 0; otherwise. (.7) sab(, ; M, N, U, V) GSABM N U ESAB M V N ESAB M N GSAB M V V (.8) Example.4: Given the following favorite probability distribution of the experiment GSAB 4 5 4. 2 4 0 0.006 0.008 0.001 1 0.091 0.122 0.015 2 0.206 0.274 0.04 0.091 0.122 0.015 4 0.006 0.008 0.001 Find the probability of getting (i) 0, 2, (ii) 2, and (iii) 4. Solution: 2 4 P() 0 0.006 0.008 0.001 0.015 1 0.091 0.122 0.015 0.228 2 0.206 0.274 0.04 0.514 0.091 0.122 0.015 0.228 4 0.006 0.008 0.001 0.015 P() 0.400 0.54 0.066 1 We get from the above table (i) Probability of getting 0, 2 i.e., P( 0, 2) 0.06 (ii) Probability of getting 2, i.e., P( 2, ) 0.274 (iii) Probability of getting 4 i.e., P( 4) 0.066. Definition.5 ASB distribution: A two dimensional random variable (, ) is said to follow ASB distribution if it assumes only non-negative values and its joint probability P(, ) asb(, ; M, N, U, V) GASBM N U EASB M V N ;,, U and V value as (.1) 0; otherwise. (.9) asb(, ; M, N, U, V) GASBM N U EASB M V N EASB M N GASB M V V (.10) Example.5: Given the following bivariate probability

American Journal of Mathematics and Statistics 2018, 8(4): 89-95 91 distribution of the experiment GASB 6 7 5. 1 2 0 0.004 0.018 0.009 1 0.04 0.172 0.086 2 0.07 0.291 0.146 0.02 0.090 0.045 Find (i) P( 2), (ii) P( 0, ), (iii) P( 2, 2) and (iv) P(, 2). Solution: 1 2 P() 0 0.004 0.018 0.009 0.01 1 0.04 0.172 0.086 0.01 2 0.07 0.291 0.146 0.510 0.02 0.090 0.045 0.158 P() 0.14 0.571 0.286 1 Solution: We get from the above table (i) P( 2) P( 2) + P( ) 0.510 + 0.158 0.668 (ii) P( 0, ) 0.009 (iii) P( 2, 2) P( 0, 2) + P(, 2) + P( 2, 2) 0.018 + 0.172 + 0.291 0.481 (iv) P(, 2) P(, 2) + P(, ) 0.090 + 0.045 0.15 Now we describe three kinds of B distributions i.e., first way B distribution, second way B distribution and both way B distribution and their different forms. Definition.6 First way B distribution: Let the joint distribution of two random variables and is given then the first way B distribution can be written as P() b 1 (; M, N, U, V) GB 1 M V N U EB M V N ;, U and V value as (.1) 0; otherwise. (.11) P() b 1 (; M, N, U, V) 1 i.e., GB 1 M V N U EB M N V EB M N V GB 1 M (.12) The first way B distribution is also known as the marginal probability distribution of. Example.6: Given the following bivariate probability distribution of example.4 find first way B distribution of (i) 0 and (ii) 2. Solution: From the given table we get (i) P( 0) 0.015 and (ii) P( 2) 0.514. Definition.7 Second way B distribution: Let the joint distribution of two random variables and is given then the second way B distribution can be written as P() b 2 (; M, N, U, V) GB 2 M V N U EB M V N ;, U and V value as (.1) 0; otherwise. (.1) P() b 2 (; M, N, U, V) i.e., GB 2 M V N U EB M N V EB M N V GB 2 M (.14) The second way B distribution is also known as the marginal probability distribution of. Example.7: Given the following bivariate probability distribution of example.4 find second way B distribution of (i) 2 and (ii) 4. Solution: From the given table we get (i) P( 2) 0.400 and (ii) P( 4) 0.066. Definition.8 Both way B distribution: A two dimensional random variable (, ) is said to follow both way B distribution if it assumes only non-negative values P(, ) b 12 (, ; M, N, U, V) GB 12 M N U EB M V N ;,, U and V value as (.1) 0; otherwise. (.15) b 12 (, ; M, N, U, V) i.e., GB 12 M N U EB M V N EB M N V GB 12 M V (.16) The both way B distribution is also known as the joint probability distribution of and. Example.8: Let the bivariate probability distribution of example.4 find both way B distribution of (i) P( 0, 4) and (ii) P( 4, ). Solution: From the given table we get (i) P( 0, 4) 0.001 and (ii) P( 4, ) 0.008. Definition.9 First way SB distribution: Let the joint distribution of two random variables and is given then the first way SB distribution can be written as P() sb 1 (; M, N, U, V) GSB 1 M V N U ESB M V N ;, U and V value as (.1) 0; otherwise. (.17)

92 Deapon Biswas: Biswas Distribution P() sb 1 (; M, N, U, V) i.e., GSB 1 M V N U ESB M N V ESB M N V GSB 1 M (.18) Example.9: Consider the GSB experiment GSB 1 6 6. Find the probability of getting (i) and (ii) 4 2. (i) P( ) GSB 1 6 6 4 1 ESB 6 6 (ii) P( 2) GSB 1 6 6 4 2 ESB 6 6 F 6 4 1 C6 4 900 4 F 6 4 C6 4 F 6 4 2 C6 4 4 675 F 6 4 C6 4 1890 0.476 1890 0.57. Definition.10 Second way SB distribution: Let the then the second way SB distribution can be written as P() sb 2 (; M, N, U, V) GSB 2 M V N U ESB M V N ;, U and V value as (.1) 0; otherwise. (.19) P() sb 2 (; M, N, U, V) i.e., GSB 2 M V N U ESB M N V ESB M N V GSB 2 M (.20) Example.10: Consider the GSB experiment GSB 2 6 6. Find the probability of getting 4 (i) and (ii). (i) P(1) GSB 2 6 6 4 1 ESB 6 6 (ii) P(2) GSB 2 6 6 4 1 ESB 6 6 F 6 4 C6 4 1 4 F 6 78 4 C6 4 F 6 4 C6 4 4 F 6 78 4 C6 4 1890 0.2 1890 0.2. Definition.11 Both way SB distribution: A two dimensional random variable (, ) is said to follow both way SB distribution if it assumes only non-negative values P(, ) sb 12 (, ; M, N, U, V) GSB 12 M V ESB M N V ;,, U and V value as (.1) 0; otherwise. (.21) sb 12 (, ; M, N, U, V) i.e., GSB 12 M N U ESB M V N ESB M N V GSB 12 M V (.22) Example.11: Consider the GSB experiment GSB 12 6 6. Find the probability of getting (i) 0, 2 4 and (ii),. (i) P( 0, 2) GSB 12 6 4 6 0 2 ESB 6 6 0.071 (ii) P(, ) GSB 12 6 4 6 ESB 6 6 0.009. 4 F 4 F 6 4 0 C6 4 2 F 6 5 4 C6 4 1890 6 4 C6 4 F 6 4 C6 4 18 1890 Definition.12 First way AB distribution: Let the joint distribution of two random variables and is given then the first way AB distribution can be written as P() ab 1 (; M, N, U, V) GAB 1 M V N U EAB M V N ;, U and V value as (.1) 0; otherwise. (.2) P() ab 1 (; M, N, U, V) i.e., GAB 1 M V N U EAB M N V EAB M N V GAB 1 M (.24) Example.12: Consider the GAB experiment GAB 1 4 5 4. Find the probability of getting (i) and (ii). (i) P(1) GAB 1 4 5 4 1 EAB 4 5 (ii) P() GAB 1 4 5 4 EAB 4 5 H 4 4 1 P5 H 4 240 P5 H 4 4 P5 H 4 440 P5 840 0.062 840 0.75. Definition.1 Second way AB distribution: Let the then the second way AB distribution can be written as P() ab 2 (; M, N, U, V) GAB 2 M V N U EAB M V N ;, U and V value as (.1) 0; otherwise. (.25) P() ab 2 (; M, N, U, V) i.e., GAB 2 M V N U EAB M N V EAB M N V GAB 2 M (.26) Example.1: Consider the GAB experiment GAB 2 4 5 4. Find the probability of getting (i) 2 and (ii)

American Journal of Mathematics and Statistics 2018, 8(4): 89-95 9. (i) P(2) GAB 2 4 5 2 4 EAB 4 5 (ii) P() GAB 2 4 5 4 EAB 4 5 H 4 P5 4 2 204 H 4 P5 H 4 P5 4 156 H 4 P5 840 0.6. 840 0.4. Definition.14 Both way AB distribution: A two dimensional random variable (, ) is said to follow both way AB distribution if it assumes only non-negative values P(, ) ab 12 (, ; M, N, U, V) GAB 12 M N U EAB M V N ;,, U and V value as (.1) 0; otherwise. (.27) ab 12 (, ; M, N, U, V) i.e., GAB 12 M N U EAB M V N EAB M N V GAB 12 M V (.28) Example.14: Consider the GAB experiment GAB 12 4 5 4. Find the probability of getting (i), 2 and (ii),. (i) P(, 2) GAB 12 4 1 5 2 4 EAB 4 5 0.07 (ii) P(, ) GAB 12 4 5 4 EAB 4 5 0.15. F 4 4 1 C5 4 2 H 4 44 P5 840 F 4 4 C5 4 H 4 576 P5 840 Definition.15 First way SAB distribution: Let the then the first way SAB distribution can be written as P() sab 1 (; M, N, U, V) GSAB 1 M V N U ESAB M V N ;, U and V value as (.1) 0; otherwise. (.29) P() sab 1 (; M, N, U, V) i.e., GSAB 1 M V N U ESAB M N V ESAB M N V GSAB 1 M (.0) Example.15: Consider the GSAB experiment GSAB 1 4 5 4. Find the probability of getting (i) 2 and (ii) 1. Solution: The table of example 27..4 represents bivatrate probability distribution of the experiment GSAB 1 4 5 4. Now we get from the table (i) P( 2) P( 2) + P( ) + P( 4) 0.514 + 0.228 + 0.015 0.757 (ii) P( 1) P( 0) + P( ) 0.015 + 0.228 0.24. Definition.16 Second way SAB distribution: Let the then the second way SAB distribution can be written as P() sab 2 (; M, N, U, V) GSAB 2 M V N U ESAB M V N ;, U and V value as (.1) 0; otherwise. (.1) P() sab 2 (; M, N, U, V) i.e., GSAB 2 M V N U ESAB M N V ESAB M N V GSAB 2 M (.2) Example.16: Consider the GSAB experiment GSAB 2 4 5 4. Find the probability of (i) and (ii). Solution: The table of example 27..4 represents GSAB 2 4 5 4. Now we get from the table (i) P ( ) 0.54 (ii) P ( ) P( 2) + P( ) 0.400 + 0.54 0.94. Definition.17 Both way SAB distribution: A two dimensional random variable (, ) is said to follow both way SAB distribution if it assumes only non- negative values P(, ) sab 12 (, ; M, N, U, V) GSAB 12 M N U ESAB M V N ;,, U and V value as (.1) 0; otherwise. (.) sab 12 (, ; M, N, U, V) i.e., GSAB 12 M N U ESAB M V N ESAB M N V GSAB 12 M V (.4) Example.17: Consider the GSAB experiment GSAB 12 4 5 4. Find the probability of getting (i), and (ii) 2, 2. Solution: The table of example 27..4 represents GSAB 12 4 5 4. Now we get from the table x y (i) P(, ) 0.122

94 Deapon Biswas: Biswas Distribution (ii) P( 2, 2) 0.206. Definition.18 First way ASB distribution: Let the then the first way ASB distribution can be written as P() asb 1 (; M, N, U, V) GASB 1 M V N U EASB M V N ;, U and V value as (.1) 0; otherwise. (.5) P() asb 1 (; M, N, U, V) i.e., GASB 1 M V N U EASB M N V EASB M N V GASB 1 M (.6) Example.18: Consider the GASB experiment GASB 1 6 7. Find the probability of getting (i) 0 and 5 (ii). Solution: The table of example 27..5 represents GASB 1 6 7. Now we get from the table 5 (i) P( 0) 0.01 (ii) P( ) 0.158. Definition.19 Second way ASB distribution: Let the then the second way ASB distribution can be written as P() asb 2 (; M, N, U, V) GASB 2 M V N U EASB M V N ;, U and V value as (.1) 0; otherwise. (.7) P() asb 2 (; M, N, U, V) i.e., GASB 2 M V N U EASB M N V EASB M N V GASB 2 M (.8) Example.19: Consider the GASB experiment GASB 2 6 7. Find the probability of getting (i) 2 and 5 (ii) 2. Solution: The table of example 27..5 represents bi-vatrate probability distribution of the experiment GASB 2 6 7. Now we get from the table 5 (i) P( 2) 0.571 (ii) P( 2) P( ) + P( 2) 0.14 + 0.571 0.714. Definition.20 Both way ASB distribution: A two dimensional random variable (, ) is said to follow both way ASB distribution if it assumes only non negative values P(, ) asb 12 (, ; M, N, U, V) GASB 12 M N U EASB M V N ;,, U and V value as (.1) 0; otherwise. (.9) asb 12 (, ; M, N, U, V) i.e., GASB 12 M N U EASB M V N EASB M N V GASB 12 M V (.40) Example.20: Consider the GASB experiment GASB 12 6 7. Find the probability of getting (i) 0, 5 2 and (ii) 2,. Solution: The table of example 27..5 represents GASB 12 6 7. Now we get from the table 5 (i) P( 0, 2) 0.018 (ii) P( 2, ) 0.146.. Applications From the definitions stated in this paper we can find various joint probability functions, marginal probability functions and conditional probability functions. 4. Main Results at a Glance The following is a list of probability laws developed in this paper. (i) b(, ; M, N, U, V) GBM N U EB M V N (ii) sb(, ; M, N, U, V) GSBM N U ESB M V N (iii) ab(, ; M, N, U, V) GABM N U EAB M V N (iv) sab(, ; M, N, U, V) GSABM N U ESAB M V N (v) asb(, ; M, N, U, V) GASBM N U EASB M V N (vi) b 1 (; M, N, U, V) GB 1 M V N U EB M V N (vii) b 2 (; M, N, U, V) GB 2 M V N U EB M V N (viii) b 12 (, ; M, N, U, V) GB 12 M N U EB M V N (ix) sb 1 (; M, N, U, V) GSB 1 M V N U ESB M V N (x) sb 2 (; M, N, U, V) GSB 2 M V N U ESB M V N

American Journal of Mathematics and Statistics 2018, 8(4): 89-95 95 (xi) sb 12 (, ; M, N, U, V) GSB 12 M V ESB M N V (xii) ab 1 (; M, N, U, V) GAB 1 M EAB M N V (xiii) ab 2 (; M, N, U, V) GAB 2 M EAB M N V (xiv) ab 12 (, ; M, N, U, V) GAB 12 M V EAB M N V (xv) sab 1 (; M, N, U, V) GSAB 1 M ESAB M N V (xvi) sab 2 (; M, N, U, V) GSAB 2 M ESAB M N V (xvii) sab 12 (, ; M, N, U, V) GSAB 12 M V ESAB M N V (xviii) asb 1 (; M, N, U, V) GASB 1 M EASB M N V (xix) asb 2 (; M, N, U, V) GASB 2 M EASB M N V (xx) asb 12 (, ; M, N, U, V) GASB 12 M V EASB M N V 5. Glossary b(, ; M, N, U, V) Biswas distribution sb(, ; M, N, U, V) SB distribution ab(, ; M, N, U, V) AB distribution sab(, ; M, N, U, V) SAB distribution asb(, ; M, N, U, V) ASB distribution b 1 (; M, N, U, V) First way B distribution b 2 (; M, N, U, V) Second way B distribution b 12 (, ; M, N, U, V) Both way B distribution sb 1 (; M, N, U, V) First way SB distribution sb 2 (; M, N, U, V) Second way SB distribution sb 12 (, ; M, N, U, V) Both way SB distribution ab 1 (; M, N, U, V) First way AB distribution ab 2 (; M, N, U, V) Second way AB distribution ab 12 (, ; M, N, U, V) Both way AB distribution sab 1 (; M, N, U, V) First way SAB distribution sab 2 (; M, N, U, V) Second way SAB distribution sab 12 (, ; M, N, U, V) Both way SAB distribution asb 1 (; M, N, U, V) First ay ASB distribution asb 2 (; M, N, U, V) Second way ASB distribution asb 12 (, ; M, N, U, V) Both way ASB distribution 6. Conclusions The paper finds an attractive Biswas distribution and related distributions. We get various joint probability functions, marginal probability functions and conditional probability functions from Biswas distribution. Combination distribution, permutation distribution, formation distribution and homogenation distribution are also found here. REFERENCES [1] Deapon Biswas, Paper 4, B space, Bystematics My Classic, 2010 Self published, Chittagong, 2016 Monon Prokashon, Chittagong, Bystematics Vol. I and II, My Classic, 2018 Scholar s Press EU, ISBN: 987-620-2-0664-5 and 987-620-2-0960-8. [2] Deapon Biswas, Paper 1, On the combinations, Bystematics My [] Deapon Biswas, Paper 16, On the permutations, Bystematics My [4] Deapon Biswas, Paper 18, Fomrations, Bystematics My [5] Deapon Biswas, Paper 20, Homogenations, Bystematics My [6] Deapon Biswas, Paper 2, EB members, Bystematics My [7] Deapon Biswas, Paper 24, IB members, Bystematics My [8] Deapon Biswas, Paper 25, CB members, Bystematics My [9] Deapon Biswas, Paper 26, GB members, Bystematics My

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