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 3.1 Biswas distribution: A two dimensional random variable (X, Y) is said to follow Biswas distribution if it assumes only non negative values and its joint probability mass function is given by

(3.1)

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 X takes only non-negative values under the interval min(M, U)−M +1 ≤ X ≤ min(M, U, V) and Y takes only non-negative values under the interval min(N, U)−N+V ≤ Y ≤ min(N, U, V). Any two dimensional random variable which follows B distribution is known as B-variate and denoted by the symbol (X, Y) ∼ B(M, N, U, V).

Remark: It should be noted that

(3.2)

Example 3.1: For the GB experiment find P(X = 2, Y = 1) where the experiment is first way arranged and second way selected.

Solution: We know

Definition 3.2 SB distribution: A two dimensional random variable (X, Y) is said to follow SB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.3)

Remark: It should be noted that

(3.4)

Example 3.2: For the GSB experiment find (i) P(X =2, Y=3) and (ii) P(X=3,Y=1).

Solution: We get

Definition 3.3 AB distribution: A two dimensional random variable (X, Y) is said to follow AB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.5)

Remark: It should be noted that

(3.6)

Example 3.3: For the GAB experiment find (i) P(X=1,Y=3) and (ii) P(X=2,Y=2).

Solution: We get

Definition 3.4 SAB distribution: A two dimensional random variable (X, Y) is said to follow SAB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.7)

Remark: It should be noted that

(3.8)

Example 3.4: Given the following favorite probability distribution of the experiment .

Find the probability of getting (i) X = 0, Y = 2, (ii) X = 2, Y = 3 and (iii) Y = 4.

Solution:

We get from the above table

(i) Probability of getting X = 0, Y = 2 i.e., P(X = 0, Y = 2) = 0.06

(ii) Probability of getting X = 2, Y = 3 i.e., P(X = 2, Y = 3) = 0.274

(iii) Probability of getting Y = 4 i.e., P(Y = 4) = 0.066.

Definition 3.5 ASB distribution: A two dimensional random variable (X, Y) is said to follow ASB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.9)

Remark: It should be noted that

(3.10)

Example 3.5: Given the following bivariate probability distribution of the experiment .

Find (i) P(X ≥ 2), (ii) P(X = 0, Y = 3), (iii) P(X ≤ 2, Y = 2) and (iv) P(X = 3, Y ≥ 2).

Solution:

Solution: We get from the above table

(i) P(X ≥ 2) = P(X = 2) + P(X = 3) = 0.510 + 0.158 = 0.668

(ii) P(X = 0, Y = 3) = 0.009

(iii) P(X ≤ 2, Y = 2) = P(X = 0, Y = 2) + P(X = 1, Y = 2) + P(X= 2, Y= 2) = 0.018 + 0.172 + 0.291 = 0.481

(iv) P(X = 3, Y ≥ 2) = P(X = 3, Y = 2) + P(X = 3, Y = 3) = 0.090 + 0.045= 0.135

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 3.6 First way B distribution: Let the joint distribution of two random variables X and Y is given then the first way B distribution can be written as

(3.11)

Remark: It should be noted that

(3.12)

The first way B distribution is also known as the marginal probability distribution of X.

Example 3.6: Given the following bivariate probability distribution of example 3.4 find first way B distribution of (i) X = 0 and (ii) X = 2.

Solution: From the given table we get (i) P(X = 0) = 0.015 and (ii) P(X = 2) = 0.514.

Definition 3.7 Second way B distribution: Let the joint distribution of two random variables X and Y is given then the second way B distribution can be written as

(3.13)

Remark: It should be noted that

(3.14)

The second way B distribution is also known as the marginal probability distribution of Y.

Example 3.7: Given the following bivariate probability distribution of example 3.4 find second way B distribution of (i) Y = 2 and (ii) Y = 4.

Solution: From the given table we get (i) P(Y = 2) = 0.400 and (ii) P(Y = 4) = 0.066.

Definition 3.8 Both way B distribution: A two dimensional random variable (X, Y) is said to follow both way B distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.15)

Remark: It should be noted that

(3.16)

The both way B distribution is also known as the joint probability distribution of X and Y.

Example 3.8: Let the bivariate probability distribution of example 3.4 find both way B distribution of (i) P(X = 0, Y = 4) and (ii) P(X = 4 , Y = 3).

Solution: From the given table we get (i) P(X = 0, Y= 4) = 0.001 and (ii) P(X = 4, Y = 3) = 0.008.

Definition 3.9 First way SB distribution: Let the joint distribution of two random variables X and Y is given then the first way SB distribution can be written as

(3.17)

Remark: It should be noted that

(3.18)

Example 3.9: Consider the GSB experiment . Find the probability of getting (i) X = 1 and (ii) X = 2.

Solution: We get the probabilities as

Definition 3.10 Second way SB distribution: Let the joint distribution of two random variables X and Y is given then the second way SB distribution can be written as

(3.19)

Remark: It should be noted that

(3.20)

Example 3.10: Consider the GSB experiment . Find the probability of getting

(i) Y = 1 and (ii) Y = 3.

Solution: We get the probabilities as

Definition 3.11 Both way SB distribution: A two dimensional random variable (X, Y) is said to follow both way SB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.21)

Remark: It should be noted that

(3.22)

Example 3.11: Consider the GSB experiment . Find the probability of getting (i) X = 0, Y = 2 and (ii) X = 3, Y = 3.

Solution: We get the probabilities as

Definition 3.12 First way AB distribution: Let the joint distribution of two random variables X and Y is given then the first way AB distribution can be written as

(3.23)

Remark: It should be noted that

(3.24)

Example 3.12: Consider the GAB experiment . Find the probability of getting (i) X = 1 and (ii) X = 3.

Solution: We get the probabilities as

Definition 3.13 Second way AB distribution: Let the joint distribution of two random variables X and Y is given then the second way AB distribution can be written as

(3.25)

Remark: It should be noted that

(3.26)

Example 3.13: Consider the GAB experiment . Find the probability of getting (i) Y = 2 and (ii) Y = 3.

Solution: We get the probabilities as

Definition 3.14 Both way AB distribution: A two dimensional random variable (X, Y) is said to follow both way AB distribution if it assumes only non-negative values and its joint probability mass function is given by

(3.27)

Remark: It should be noted that

(3.28)

Example 3.14: Consider the GAB experiment . Find the probability of getting (i) X = 1, Y = 2 and (ii) X = 3, Y = 3.

Solution: We get the probabilities as

Definition 3.15 First way SAB distribution: Let the joint distribution of two random variables X and Y is given then the first way SAB distribution can be written as

(3.29)

Remark: It should be noted that

(3.30)

Example 3.15: Consider the GSAB experiment . Find the probability of getting (i) X ≥ 2 and (ii) X ≤ 1.

Solution: The table of example 27.3.4 represents bivatrate probability distribution of the experiment . Now we get from the table

(i) P(X ≥ 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.514 + 0.228 + 0.015 = 0.757

(ii) P(X ≤ 1) = P(X = 0) + P(X = 1) = 0.015 + 0.228 = 0.243.

Definition 3.16 Second way SAB distribution: Let the joint distribution of two random variables X and Y is given then the second way SAB distribution can be written as

(3.31)

Remark: It should be noted that

(3.32)

Example 3.16: Consider the GSAB experiment . Find the probability of (i) Y = 3 and (ii) Y ≤ 3.

Solution: The table of example 27.3.4 represents bi-variate probability distribution of the experiment . Now we get from the table

(i) P (Y = 3 ) = 0.534

(ii) P (Y ≤ 3) = P(Y = 2) + P(Y = 3) = 0.400 + 0.534 = 0.934.

Definition 3.17 Both way SAB distribution: A two dimensional random variable (X, Y) is said to follow both way SAB distribution if it assumes only non- negative values and its joint probability mass function is given by

(3.33)

Remark: It should be noted that

(3.34)

Example 3.17: Consider the GSAB experiment . Find the probability of getting

(i) X = 1, Y = 3 and (ii) X = 2, Y = 2.

Solution: The table of example 27.3.4 represents bi-variate probability distribution of the experiment . Now we get from the table

(i) P(X = 1, Y = 3) = 0.122

(ii) P(X = 2, Y = 2) = 0.206.

Definition 3.18 First way ASB distribution: Let the joint distribution of two random variables X and Y is given then the first way ASB distribution can be written as

(3.35)

Remark: It should be noted that

(3.36)

Example 3.18: Consider the GASB experiment . Find the probability of getting (i) X = 0 and (ii) X = 3.

Solution: The table of example 27.3.5 represents bi-variate probability distribution of the experiment . Now we get from the table

(i) P(X = 0) = 0.031

(ii) P(X = 3) = 0.158.

Definition 3.19 Second way ASB distribution: Let the joint distribution of two random variables X and Y is given then the second way ASB distribution can be written as

(3.37)

Remark: It should be noted that

(3.38)

Example 3.19: Consider the GASB experiment . Find the probability of getting (i) Y = 2 and (ii) Y ≤ 2.

Solution: The table of example 27.3.5 represents bi-vatrate probability distribution of the experiment . Now we get from the table

(i) P(Y = 2) = 0.571

(ii) P(Y ≤ 2) = P(Y = 1) + P(Y = 2) = 0.143 + 0.571= 0.714.

Definition 3.20 Both way ASB distribution: A two dimensional random variable (X, Y) is said to follow both way ASB distribution if it assumes only non negative values and its joint probability mass function is given by

(3.39)

Remark: It should be noted that

(3.40)

Example 3.20: Consider the GASB experiment . Find the probability of getting (i) X = 0, Y = 2 and (ii) X = 2, Y = 3.

Solution: The table of example 27.3.5 represents bi-variate probability distribution of the experiment . Now we get from the table

(i) P(X = 0, Y = 2) = 0.018

(ii) P(X = 2, Y = 3) = 0.146.

3. 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.

5. Glossary

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.