# Bivariate Discrete Distribution

12 de Dec de 2021
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### Bivariate Discrete Distribution

• 1. Bivariate Discrete Distribution Presented By Arijit Dhali Ipsita Raha
• 3. ABSTRACT Bivariate Discrete Distributions details the latest techniques of computer simulation for the distributions considered. It contains a general introduction to the structural properties of discrete distributions, including generating functions, moment relationships, and the basic ideas of generalizing and much more.
• 4. KEYWORDS 01 Bivariate Data Joint Probability 03 Marginal Distribution 02 Marginal probability Mass 04
• 5. INTRODUCTION In this presentation we will consider two or more random variables defined on the same sample space and discuss how to model the probability distribution of the random variables jointly. We will begin with the discrete case by looking at the joint probability mass function for two discrete random variables.
• 6. DISCUSSION JOINT DISTRIBUTION OF RANDOM BIVARIATES Let X be a random variable the values X : x1 x2 x3 ……. xm Let Y be a random variable assuming the following values corresponding to each xi Y : y1 y2 y3 ……. Yn In the table that will be shown in next slide, There are mn number of values (xi,yj) . These values are known as Bivariate Data. Also, Pij = Probability of assuming the pair (xi,yj) by (X,Y) = P { (xi,yj) } = P ( X = xi, Y = yj ) are known as Joint Probability Mass of Bivariate (X,Y)
• 7. BIVARIATE JOINT DISTRIBUTION TABLE Where the row wise and column wise total are: • PXi = pi1 + pi2 + …. + pin = σ𝑗=1 𝑛 𝑝𝑖𝑗 • PYi = p1j + p2j + …. + pmj = σ𝑖=1 𝑚 𝑝𝑖𝑗 The grand total: • σ𝑖=1 𝑚 𝑝𝑋𝑖 + σ𝑗=1 𝑛 𝑝𝑌𝑗 = σ𝑗=1 𝑛 σ𝑖=1 𝑚 𝑝𝑖𝑗 = 1
• 8. MARGINAL DISTRIBUTION This probability distribution of X is called as Marginal Distribution of X X : x1 x2 x3 …… xm Total pXi: pX1 pX2 pX3 …… pXm 1 Similarly this probability distribution of Y is called as Marginal Distribution of Y Y : y1 y2 y3 …… yn Total pYj: pY1 pY2 pY3 …… pYn 1 The row wise totals PXi and the column wise totals PYi are called Marginal Probability Mass of X and Y respectively.
• 9. INDEPENDENT RANDOM VARIABLES Let (X,Y) be a pair of random variables having joint distribution discussed in previous slide. If pij = pXi pYj = P ( X = xi, Y = yj ) = P ( X = xi ) P ( Y = yj ) Hold for all values of i ( 1 ≤ i ≤ m ) and j ( 1 ≤ j ≤ n ) then X and Y are called Independent Random Variables. Theorem: If X and Y are independent random variables and A, B are two events then P { X ∈ A, Y ∈ B } = P ( X ∈ A ) P ( Y ∈ B ) and vice versa.
• 10. Problem: Let (X,Y) be a bivariate having the following joint distribution: Check whether X and Y are independent or not Solution: Here we see every data in the main body of the table is equal to the product of the corresponding data in last column and last row, e.g 0.35 = 0.70 x 0.50, 0.06 = 0.30 x 0.20 etc. ILLUSTRATIVE EXAMPLE 1 That is P ( X = 0.20, Y = 5 ) = P ( X = 0.20 ) P ( Y = 5), P ( X = 9, Y = 7 ) = P ( X = 9 ) P ( Y = 7 ) etc. So, X & Y are independent random variable.
• 11. Problem: An urn contains 3 Red, 2 White and 5 Blue balls. Three balls are drawn from the urn. X and Y denote the number of Red and White balls in a draw. Find the Joint Distribution of (X,Y). Hence find P ( X ≤ 2, Y ≥ 1 ). Find the Marginal Distribution of Y and hence find the probability of drawing more than 1 White balls. Are X and Y independent random variable? Solution: Consider X and Y as Probability of Red and White balls. Take the values of X as 0, 1, 2, 3 and Y as 0, 1, 2. Create a table with values of the bivariate (X,Y). Where, P { (xi,yj) } = P ( X = xi, Y = yj ) The corresponding probabilities are: 1) P(0,0) = Probability of “no Red”,”no White”,”3 Blue” = Τ 5 𝐶3 10 𝐶3 = 0.083 ILLUSTRATIVE EXAMPLE 2
• 12. 2) P(0,1) = Probability of “no Red”,”1 White”,”2 Blue” = Τ 2 𝐶1 ∗5 𝐶2 10 𝐶3 = 0.16 3) P(0,2) = Probability of “no Red”,”2 White”,”1 Blue” = Τ 2 𝐶2 ∗5 𝐶1 10 𝐶3 = 0.041 4) P(1,0) = Probability of “1 Red”,”no White”,”2 Blue” = Τ 3 𝐶1 ∗5 𝐶2 10 𝐶3 = 0.25 5) P(1,1) = Probability of “1 Red”,”1 White”,”1 Blue” = Τ 3 𝐶1 ∗2 𝐶1 ∗5 𝐶1 10 𝐶3 = 0.25 6) P(1,2) = Probability of “1 Red”,”2 White”,”no Blue” = Τ 3 𝐶1 ∗2 𝐶2 10 𝐶3 = 0.025 7) P(2,0) = Probability of “2 Red”,”no White”,”1 Blue” = Τ 3 𝐶2 ∗5 𝐶1 10 𝐶3 = 0.125 8) P(2,1) = Probability of “2 Red”,”1 White”,”no Blue” = Τ 3 𝐶2 ∗2 𝐶1 10 𝐶3 = 0.05 9) P(2,2) = Probability of “2 Red”,”2 White”,”no Blue” = P(ϕ) = 0 10) P(3,0) = Probability of “3 Red”,”no White”,”no Blue” = Τ 3 𝐶3 10 𝐶3 = 0.008 11) P(3,1) = Probability of “3 Red”,”1 White”,”no Blue” = P(ϕ) = 0 12) P(3,2) = Probability of “3 Red”,”2 White”,”no Blue” = P(ϕ) = 0 ILLUSTRATIVE EXAMPLE 2 - CONTINUED }
• 13. ∴ The Joint Distribution of (X,Y) is given by: Now, P (X ≤ 2, Y ≥ 1 ) = P ( 2,1 ) + P ( 2,2 ) + P ( 1,1 ) + P ( 1,2 ) + P ( 0,1 ) + P ( 0,2 ) = 0.05 + 0 + 0.25 + 0.025 + 0.16 + 0.041 = 0.526 The Marginal Distribution of Y is given by Y : 0 1 2 pYj : 0.466 0.466 0.066 ILLUSTRATIVE EXAMPLE 2 - CONTINUED Probability of “ more than 1 White balls “: P ( Y ≥ 1 ) = 0.466 + 0.133 = 0.6 From the above table we see, = 0.083 ≠ 0.284 x 0.466 ∴ X, Y are not independent
• 14. APPLICATIONS The bivariate distribution is useful in analyzing the relationship between two randomly distributed variables, and thus has heavy application to biology and economics where the relationship between approximately-random variables is of great interest. For instance, one of the earliest uses of the bivariate distribution was in analyzing the relationship between a father's height and the height of their eldest son, resolving a question Darwin posed in his book the “The Origin of Species”. Also used in measuring systems, such as those used in coordinate measuring machines (CMMs), laser interferometers, linear or rotary encoders, etc.
• 15. CONCLUSION In real life, we are often interested in several random variables that are related to each other. For example, suppose that we choose a random family, and we would like to study the number of people in the family, the household income, the ages of the family members, etc. Each of these is a random variable, and we suspect that they are dependent. In this presentation, we developed the tools to study joint distributions of random variables.
• 16. REFERENCES • [1]https://online.stat.psu.edu/stat414/lesson/17/17.1 • [2]https://bookdown.org/compfinezbook/introcompfinr/Bivariate- Distributions.html • [3]https://en.wikipedia.org/wiki/Random_variable#Discrete_random _variable • [4]https://en.wikipedia.org/wiki/Joint_probability_distribution • [5]https://www.probabilitycourse.com/chapter5/5_1_0_joint_distribut ions.php • [6]https://www.stat.ncsu.edu/people/bloomfield/courses/st380/slide s/Devore-ch05-sec1-2.pdf • [7]https://brilliant.org/wiki/multivariate-normal-distribution/ • [8]Page 128 Engineering Mathematics Vol – 2A by B.K.Pal & K.Das, published by U.N Dhur & Sons Private Ltd.
• 17. THANK YOU For listening patiently