Two Dimensional Discrete Random Variables play a crucial role in the study of Probability and Statistics. Their importance lies in modeling the relationship between two random variables. This topic is essential in understanding joint distributions, covariance, and correlation.
Introduction #
Two Dimensional Discrete Random Variables are essential concepts in Probability and Statistics. They provide a method to model the relationship between two discrete random variables, allowing for the understanding of their joint behavior. The study of these variables helps in calculating joint probabilities, expectations, and in understanding how two random variables interact.
Discrete Random Variables #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\). Let \(X\) and \(Y\) be random variables on \(S\). The joint distribution function \(F(x,y)\) of a random variable \(X,Y\) is said to be discrete if the spectrum of \(X,Y\) is at most countable.
Let the spectrum of \(X,Y\) contains the points \((x_{i},y_{j}),~i,j=0,\pm 1,\pm 2,… \) such that \(…,x_{-2},x_{-1},x_{0},x_{1},x_{2},.. \) and \(…,y_{-2},y_{-1},y_{0},y_{1},y_{2},.. \) and \(f_{ij}=P(X=x_{i},Y=y_{i}) \) with \( \displaystyle\sum_{j=-\infty}^{\infty}\displaystyle\sum_{i=-\infty}^{\infty}f_{ij}=1 \).
Then \( F(x,y)= \displaystyle\sum_{\beta=-\infty}^{j}\displaystyle\sum_{\alpha=-\infty}^{i}f_{\alpha\beta} \) where \( x_{i}\leq x \lt x_{i+1} \) and \( y_{j}\leq y \lt x_{j+1} \).
Properties #
- \( P(a\lt X \leq b,c\lt Y \leq d)=\displaystyle\sum_{c\lt y_{j} \leq d}^{}\displaystyle\sum_{a\lt x_{i} \leq b}^{}f_{ij} \)
Let \(a\lt b\) and \(c\lt d\).
Then \( \exists \) two distinct step points \( x_{s} \), \( x_{t} \) and \( y_{u} \) and \( y_{v} \) such that \( x_{s} \leq a \lt b \lt x_{t} \) and \( y_{s} \leq c \lt d \lt y_{v} \).
\(\implies x_{s} \leq a \lt x_{s+1} \leq x_{t-1} \leq b \lt x_{t} \) and \( y_{u} \leq c \lt y_{u+1} \leq y_{v-1} \leq d \lt y_{v} \).
Now \( P(a\lt X\leq b,c\lt Y\leq d)\)
\(= F(a,c)+F(b,d)-F(a,d)-F(b,c) \)
\(= \displaystyle\sum_{\beta=-\infty}^{u}\displaystyle\sum_{\alpha=-\infty}^{s}f_{\alpha\beta} +\displaystyle\sum_{\beta=-\infty}^{v-1}\displaystyle\sum_{\alpha=-\infty}^{t-1}f_{\alpha\beta} -\displaystyle\sum_{\beta=-\infty}^{v-1}\displaystyle\sum_{\alpha=-\infty}^{s}f_{\alpha\beta}-\displaystyle\sum_{\beta=-\infty}^{u}\displaystyle\sum_{\alpha=-\infty}^{t-1}f_{\alpha\beta} \)
\(=-\displaystyle\sum_{\beta=-\infty}^{u} \Big[\displaystyle\sum_{\alpha=-\infty}^{t-1}f_{\alpha\beta} -\displaystyle\sum_{\alpha=-\infty}^{s}f_{\alpha\beta} \Big]+ \displaystyle\sum_{\beta=-\infty}^{v-1}\Big[\displaystyle\sum_{\alpha=-\infty}^{t-1}f_{\alpha\beta}- \displaystyle\sum_{\alpha=-\infty}^{s}f_{\alpha\beta} \Big] \)
\(=-\displaystyle\sum_{\beta=-\infty}^{u} \displaystyle\sum_{\alpha=s}^{t-1}f_{\alpha\beta} + \displaystyle\sum_{\beta=-\infty}^{v-1}\displaystyle\sum_{\alpha=s}^{t-1}f_{\alpha\beta} \)
\(=\displaystyle\sum_{\alpha=s}^{t-1}\Big[\displaystyle\sum_{\beta=-\infty}^{v-1}-\displaystyle\sum_{\beta=-\infty}^{u} f_{\alpha\beta} \Big] \)
\(=\displaystyle\sum_{\alpha=s}^{t-1}\displaystyle\sum_{\beta=u}^{v-1}f_{\alpha\beta} \)
\(=\displaystyle\sum_{\beta=u}^{v-1}\displaystyle\sum_{\alpha=s}^{t-1}f_{\alpha\beta} \)
\(=\displaystyle\sum_{c\lt y_{j} \leq d}^{}\displaystyle\sum_{a\lt x_{i} \leq b}^{}f_{ij} \)
- If \(P(X=x_{i})=f_{i\cdot}=f_{xi} \) then \(f_{i\cdot}=\displaystyle\sum_{j=-\infty}^{\infty}f_{ij} \)
We have \( (X=x_{i})\)
\(= (X=x_{i})S\)
\(= (X=x_{i})\displaystyle\sum_{j=-\infty}^{\infty}(Y=y_{j}) \)
\(= \displaystyle\sum_{j=-\infty}^{\infty}(X=x_{i})(Y=y_{j}) \)
\(= \displaystyle\sum_{j=-\infty}^{\infty}(X=x_{i},Y=y_{j}) \)
Since \((X=x_{i},Y=y_{j}),~j=0,\pm 1,\pm 2,… \) are all mutually exclusive events.
Then \( P(X=x_{i})=P\displaystyle\sum_{j=-\infty}^{\infty}(X=x_{i},Y=y_{j})\)
\(\implies f_{i\cdot}=\displaystyle\sum_{j=-\infty}^{\infty}P(X=x_{i},Y=y_{j})\)
\(\implies f_{i\cdot}=\displaystyle\sum_{j=-\infty}^{\infty} f_{ij} \) - \(F_{x}(x)=\displaystyle\sum_{\alpha=-\infty}^{i} f_{\alpha\cdot} \) where \( x_{i}\leq x \lt x_{i+1} \).
We have \(F_{x}(x)=F(x,\infty) \)
\(\implies F_{x}(x)=\displaystyle\sum_{\beta=-\infty}^{\infty}\displaystyle\sum_{\alpha=-\infty}^{i}f_{\alpha\beta}\)
\(\implies F_{x}(x)=\displaystyle\sum_{\alpha=-\infty}^{i}\displaystyle\sum_{\beta=-\infty}^{\infty}f_{\alpha\beta}\)
\(\implies F_{x}(x)=\displaystyle\sum_{\alpha=-\infty}^{i} f_{\alpha\cdot} \)
- If \(P(Y=y_{i})=f_{\cdot j}=f_{yj} \) then \(f_{\cdot j}=\displaystyle\sum_{i=-\infty}^{\infty}f_{ij} \)
We have \( (Y=y_{i})\)
\(= S(Y=y_{i})\)
\(= \Big[\displaystyle\sum_{i=-\infty}^{\infty}(X=x_{i})\Big](Y=y_{j}) \)
\(= \displaystyle\sum_{i=-\infty}^{\infty}(X=x_{i})(Y=y_{j}) \)
\(= \displaystyle\sum_{i=-\infty}^{\infty}(X=x_{i},Y=y_{j}) \)
Since \((X=x_{i},Y=y_{j}),~i=0,\pm 1,\pm 2,… \) are all mutually exclusive events.
Then \( P(Y=y_{i})=P\displaystyle\sum_{i=-\infty}^{\infty}(X=x_{i},Y=y_{j})\)
\(\implies f_{\cdot j}=\displaystyle\sum_{i=-\infty}^{\infty}P(X=x_{i},Y=y_{j})\)
\(\implies f_{\cdot j}=\displaystyle\sum_{i=-\infty}^{\infty} f_{ij} \)
- \(F_{y}(y)=\displaystyle\sum_{\beta=-\infty}^{y} f_{\cdot\beta} \) where \( y_{j}\leq y \lt y_{j+1} \).
We have \(F_{y}(y)=F(\infty,y) \)
\(\implies F_{y}(y)=\displaystyle\sum_{\beta=-\infty}^{j}\displaystyle\sum_{\alpha=-\infty}^{\infty}f_{\alpha\beta}\)
\(\implies F_{y}(y)=\displaystyle\sum_{\beta=-\infty}^{y} f_{\cdot\beta} \)
Applications #
Two Dimensional Discrete Random Variables have numerous applications in various fields, especially in statistical analysis. They are widely used in analyzing systems where two related variables interact, such as in economics for modeling income distribution across two groups, in biology for analyzing the relationship between two traits, and in engineering for reliability analysis of systems.
Conclusion #
The study of Two Dimensional Discrete Random Variables is vital for understanding the behavior of multiple related random variables in Probability and Statistics. By modeling joint distributions, covariance, and correlation, a deeper insight into their interactions is gained. This knowledge is pivotal for various applications across different fields, including economics, biology, and engineering.
FAQs #
- What are Two Dimensional Discrete Random Variables?
Two Dimensional Discrete Random Variables are random variables that represent two related discrete outcomes. They help in understanding the joint distribution and correlation between two random variables. - How are Two Dimensional Discrete Random Variables used in statistics?
They are used to model the relationship between two discrete random variables, allowing for the calculation of joint probabilities, expectations, and understanding their interaction. - What is the importance of Two Dimensional Discrete Random Variables in Probability?
They are important for analyzing joint distributions and calculating how two random variables behave together, helping in various statistical applications. - Can Two Dimensional Discrete Random Variables be applied in real-world scenarios?
Yes, they are applied in various fields such as economics, biology, and engineering to model the relationship between two variables. - What is the relationship between covariance and Two Dimensional Discrete Random Variables?
Covariance measures how two related random variables change together. It is an important concept when dealing with Two Dimensional Discrete Random Variables. - What is a joint distribution in Two Dimensional Discrete Random Variables?
A joint distribution describes the probability distribution of two related random variables. It is essential for understanding how the two variables interact and their combined outcomes. - How are Two Dimensional Discrete Random Variables represented?
They are typically represented by a two-dimensional probability mass function, which outlines the probabilities of different combinations of the two random variables. - What is the role of correlation in Two Dimensional Discrete Random Variables?
Correlation quantifies the strength and direction of the relationship between two random variables, making it an important concept when studying Two Dimensional Discrete Random Variables. - What are some common methods to analyze Two Dimensional Discrete Random Variables?
Common methods include calculating joint probabilities, expectations, covariance, and correlation to understand the relationship between the two variables. - Why is the study of Two Dimensional Discrete Random Variables crucial in statistics?
Understanding Two Dimensional Discrete Random Variables is essential for analyzing the interaction between multiple random variables, which is key in various statistical modeling and decision-making processes.
