Two Dimensional Distribution Functions

  Two Dimensional Distribution Functions have been extensively studied in probability and statistics. They play a crucial role in analyzing the joint behavior of two random variables. The concept has been applied in various scientific fields, making it a fundamental topic in probability theory and statistical modeling.

Introduction #

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  In probability and statistics, Two Dimensional Distribution Functions are essential for understanding the relationships between two random variables. These functions help in deriving joint probability distributions and conditional probabilities, ensuring a comprehensive analysis of statistical dependencies. Their applications span various scientific and engineering disciplines.

Two Dimensional Distribution Function #

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  Let $E$ be a random experiment and $S$ be the event space of $E$. Let $X$ and $Y$ be two random variables on $S$. The joint distribution dunction of a random variable $X$ and $Y$ is a function , denoted by $F(x,y)$ or $F_{x,y}(x,y)$ defined by $F:\mathbb{R}\times \mathbb{R}\to [0,1]$ such that $F(x,y)=P(-\infty\lt X \leq x,-\infty\lt Y \leq y)$.

Properties #

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  Let $E$ be a random experiment and $S$ be the event space of $E$. Let $X$ and $Y$ be two random variables on $S$ and $F(x,y)$ be the joint distribution dunction of a random variable $X$ and $Y$.

1. $P(-\infty\lt X \leq a,c\lt Y \leq d)=F(a,d)-F(a,c)$ where $c\lt d$
   Let $c\lt d$.
   We have $(-\infty\lt X \leq a,-\infty\lt Y \leq d)$
   $=(-\infty\lt X \leq a,-\infty\lt Y \leq c)+(-\infty\lt X \leq a,c\lt Y \leq d)$.
   Since $(-\infty\lt X \leq a,-\infty\lt Y \leq c)$ and $(-\infty\lt X \leq a,c\lt Y \leq d)$ are mutually exclusive then
   $P(-\infty\lt X \leq a,-\infty\lt Y \leq d)$
   $=P(-\infty\lt X \leq a,-\infty\lt Y \leq c)+P(-\infty\lt X \leq a,c\lt Y \leq d)$.
   $\implies F(a,d)=F(a,c)+P(-\infty\lt X \leq a,c\lt Y \leq d)$
   $\implies P(-\infty\lt X \leq a,c\lt Y \leq d)=F(a,d)-F(a,c)$
   

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2. $P(a\lt X \leq b,-\infty\lt Y \leq d)=F(b,d)-F(a,d)$ where $a\lt b$
   Let $a\lt b$.
   We have $(-\infty\lt X \leq b,-\infty\lt Y \leq d)$
   $=(-\infty\lt X \leq a,-\infty\lt Y \leq d)+(a\lt X \leq b,-\infty\lt Y \leq d)$.
   Since $(-\infty\lt X \leq a,-\infty\lt Y \leq d)$ and $(a\lt X \leq b,-\infty\lt Y \leq d)$ are mutually exclusive then
   $P(-\infty\lt X \leq b,-\infty\lt Y \leq d)$
   $=P(-\infty\lt X \leq a,-\infty\lt Y \leq d)+P(a\lt X \leq b,-\infty\lt Y \leq d)$.
   $\implies F(b,d)=F(a,d)+P(a\lt X \leq b,-\infty\lt Y \leq d)$
   $\implies P(a\lt X \leq b,-\infty\lt Y \leq d)=F(b,d)-F(a,d)$
   

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3. $P(a\lt X \leq b,c\lt Y \leq d)=F(a,c)+F(b,d)-F(a,d)-F(b,c)$ where $a\lt b,~ c\lt d$
   Let $a\lt b,~ c\lt d$.
   We have $P(-\infty\lt X \leq b,c\lt Y \leq d)=F(b,d)-F(b,c)$
   and $(-\infty\lt X \leq b,c\lt Y \leq d)$
   $=(-\infty\lt X \leq a,c\lt Y \leq d)+(a\lt X \leq b,c\lt Y \leq d)$.
   Since $(-\infty\lt X \leq a,c\lt Y \leq d)$ and $(a\lt X \leq b,c\lt Y \leq d)$ are mutually exclusive then
   $P(-\infty\lt X \leq b,c\lt Y \leq d)$
   $=P(-\infty\lt X \leq a,c\lt Y \leq d)+P(a\lt X \leq b,c\lt Y \leq d)$.
   $\implies F(b,d)-F(b,c)=F(a,d)-F(a,c)+P(a\lt X \leq b,c\lt Y \leq d)$
   $\implies P(a\lt X \leq b,c\lt Y \leq d)=F(b,d)-F(b,c)-F(a,d)+F(a,c)$
   $\implies P(a\lt X \leq b,c\lt Y \leq d)=F(a,c)+F(b,d)-F(a,d)-F(b,c)$
   

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4. $F(-\infty,y)=0$
   Let $A_{n}$ be the sequence of events $(-\infty\lt X\leq -n)$, $n=1,2,…$ and $B$ be the event $(-\infty\lt Y\leq y)$.
   Then $A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…$.
   $\implies A_{1}B\supseteq A_{2}B\supseteq … \supseteq A_{n}B\supseteq…$
   Therefore $\lim\limits_{n\to \infty } A_{n}B=OB=O$
   $\implies P\Big(\lim\limits_{n\to \infty } A_{n}B\Big)=P(O)$
   $\implies \lim\limits_{n\to \infty } P\Big(A_{n}B\Big)=0$
   $\implies \lim\limits_{n\to \infty } P\Big((-\infty\lt X\leq -n)(-\infty\lt Y\leq y)\Big)=0$
   $\implies \lim\limits_{n\to \infty } P(-\infty\lt X\leq -n,-\infty\lt Y\leq y)=0$
   $\implies \lim\limits_{n\to \infty } F(-n,y)=0$
   $\implies F(-\infty,y)=0$
   

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5. $F(x,-\infty)=0$
   Let $B_{n}$ be the sequence of events $(-\infty\lt Y\leq -n)$, $n=1,2,…$ and $A$ be the event $(-\infty\lt X\leq x)$.
   Then $B_{1}\supseteq B_{2}\supseteq … \supseteq B_{n}\supseteq…$.
   $\implies AB_{1}\supseteq AB_{2}\supseteq … \supseteq AB_{n}\supseteq…$
   Therefore $\lim\limits_{n\to \infty } AB_{n}=AO=O$
   $\implies P\Big(\lim\limits_{n\to \infty } AB_{n}\Big)=P(O)$
   $\implies \lim\limits_{n\to \infty } P\Big(AB_{n}\Big)=0$
   $\implies \lim\limits_{n\to \infty } P\Big((-\infty\lt X\leq x)(-\infty\lt Y\leq -n)\Big)=0$
   $\implies \lim\limits_{n\to \infty } P(-\infty\lt X\leq x,-\infty\lt Y\leq -n)=0$
   $\implies \lim\limits_{n\to \infty } F(x,-n)=0$
   $\implies F(x,-\infty)=0$
   

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6. $F(\infty,\infty)=1$
   Let $A_{n}$ and $B_{n}$ be the sequences of events $(-\infty\lt X\leq n)$ and $(-\infty\lt Y\leq n)$, $n=1,2,…$ respectively.
   Then $A_{1}\subseteq A_{2}\subseteq … \subseteq A_{n}\subseteq…$.
   and $B_{1}\subseteq B_{2}\subseteq … \subseteq B_{n}\subseteq…$.
   $\implies A_{1}B_{1}\subseteq A_{2}B_{2}\subseteq … \subseteq A_{n}B_{n}\subseteq…$
   Therefore $\lim\limits_{n\to \infty } A_{n}B_{n}=SS=S$
   $\implies P\Big(\lim\limits_{n\to \infty } A_{n}B_{n}\Big)=P(S)$
   $\implies \lim\limits_{n\to \infty } P\Big(A_{n}B_{n}\Big)=1$
   $\implies \lim\limits_{n\to \infty } P\Big((-\infty\lt X\leq n)(-\infty\lt Y\leq n)\Big)=1$
   $\implies \lim\limits_{n\to \infty } P(-\infty\lt X\leq n,-\infty\lt Y\leq n)=1$
   $\implies \lim\limits_{n\to \infty } F(n,n)=1$
   $\implies F(\infty,\infty)=1$
   

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7. $P(X=a,-\infty\lt Y\leq y)=F(a,y)-F(a-0,y)$
   Let $A_{n}$ be the sequence of events $(a-\frac{1}{n}\lt X\leq a)$, $n=1,2,…$ and $B$ be the event $(-\infty\lt Y\leq y)$.
   Then $A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…$.
   $\implies A_{1}B\supseteq A_{2}B\supseteq … \supseteq A_{n}B\supseteq…$
   Therefore $\lim\limits_{n\to \infty } A_{n}B=(X=a)B$
   $\implies P\Big(\lim\limits_{n\to \infty } A_{n}B\Big)=P(X=a,-\infty\lt Y\leq y)$
   $\implies \lim\limits_{n\to \infty } P\Big(A_{n}B\Big)=P(X=a,-\infty\lt Y\leq y)$
   $\implies \lim\limits_{n\to \infty } P\Big((a-\frac{1}{n}\lt X\leq a)(-\infty\lt Y\leq y)\Big)=P(X=a,-\infty\lt Y\leq y)$
   $\implies \lim\limits_{n\to \infty } P(a-\frac{1}{n}\lt X\leq a,-\infty\lt Y\leq y)=P(X=a,-\infty\lt Y\leq y)$
   $\implies \lim\limits_{n\to \infty } \Big[F(a,y)-F(a-\frac{1}{n},y)\Big]=P(X=a,-\infty\lt Y\leq y)$
   $\implies F(a,y)-F(a-0,y)=P(X=a,-\infty\lt Y\leq y)$
   

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8. $P(-\infty\lt X\leq x,Y=c)=F(x,c)-F(x,c-0)$
   Let $B_{n}$ be the sequence of events $(c-\frac{1}{n}\lt Y\leq c)$, $n=1,2,…$ and $A$ be the event $(-\infty\lt X\leq x)$.
   Then $B_{1}\supseteq B_{2}\supseteq … \supseteq B_{n}\supseteq…$.
   $\implies AB_{1}\supseteq AB_{2}\supseteq … \supseteq AB_{n}\supseteq…$
   Therefore $\lim\limits_{n\to \infty } AB_{n}=A(Y=c)$
   $\implies P\Big(\lim\limits_{n\to \infty } AB_{n}\Big)=P(-\infty\lt X\leq x,Y=c)$
   $\implies \lim\limits_{n\to \infty } P\Big(AB_{n}\Big)=P(-\infty\lt X\leq x,Y=c)$
   $\implies \lim\limits_{n\to \infty } P\Big((-\infty\lt X\leq x)(c-\frac{1}{n}\lt Y\leq c)\Big)=P(-\infty\lt X\leq x,Y=c)$
   $\implies \lim\limits_{n\to \infty } P(-\infty\lt X\leq x,c-\frac{1}{n}\lt Y\leq c)=P(-\infty\lt X\leq x,Y=c)$
   $\implies \lim\limits_{n\to \infty } \Big[F(x,c)-F(x,c-\frac{1}{n})\Big]=P(-\infty\lt X\leq x,Y=c)$
   $\implies F(x,c)-F(x,c-0)=P(-\infty\lt X\leq x,Y=c)$
   

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9. $F(a+0,y)=F(a,y)$
   Let $A_{n}$ be the sequence of events $(a\lt X\leq a+\frac{1}{n})$, $n=1,2,…$ and $B$ be the event $(-\infty\lt Y\leq y)$.
   Then $A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…$.
   $\implies A_{1}B\supseteq A_{2}B\supseteq … \supseteq A_{n}B\supseteq…$
   Therefore $\lim\limits_{n\to \infty } A_{n}B=OB=O$
   $\implies P\Big(\lim\limits_{n\to \infty } A_{n}B\Big)=P(O)$
   $\implies \lim\limits_{n\to \infty } P\Big(A_{n}B\Big)=0$
   $\implies \lim\limits_{n\to \infty } P\Big((a\lt X\leq a+\frac{1}{n})(-\infty\lt Y\leq y)\Big)=0$
   $\implies \lim\limits_{n\to \infty } P(a\lt X\leq a+\frac{1}{n},-\infty\lt Y\leq y)=0$
   $\implies \lim\limits_{n\to \infty } \Big[F(a+\frac{1}{n},y)-F(a,y)\Big]=0$
   $\implies F(a+0,y)-F(a,y)=0$
   $\implies F(a+0,y)=F(a,y)$
   

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10. $F(x,c+0)=F(x,c)$
    Let $B_{n}$ be the sequence of events $(c\lt Y\leq c+\frac{1}{n})$, $n=1,2,…$ and $A$ be the event $(-\infty\lt X\leq x)$.
    Then $B_{1}\supseteq B_{2}\supseteq … \supseteq B_{n}\supseteq…$.
    $\implies AB_{1}\supseteq AB_{2}\supseteq … \supseteq AB_{n}\supseteq…$
    Therefore $\lim\limits_{n\to \infty } AB_{n}=AO=O$
    $\implies P\Big(\lim\limits_{n\to \infty } AB_{n}\Big)=P(O)$
    $\implies \lim\limits_{n\to \infty } P\Big(AB_{n}\Big)=0$
    $\implies \lim\limits_{n\to \infty } P\Big((-\infty\lt X\leq x)(c\lt Y\leq c+\frac{1}{n})\Big)=0$
    $\implies \lim\limits_{n\to \infty } P(-\infty\lt X\leq x,c\lt Y\leq c+\frac{1}{n})=0$
    $\implies \lim\limits_{n\to \infty } \Big[F(x,c+\frac{1}{n})-F(x,c)\Big]=0$
    $\implies F(x,c+0)-F(x,c)=0$
    $\implies F(x,c+0)=F(x,c)$
    

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11. $P(X=a,c\lt Y\leq d)=F(a,d)+F(a-0,c)-F(a-0,d) -F(a,c)$ where $c\lt d$
    Let $c\lt d$. We have $F(a,y)-F(a-0,y)=P(X=a,-\infty\lt Y\leq y)$
    Also $(X=a,-\infty\lt Y\leq d) =(X=a,-\infty\lt Y\leq c)+(X=a,c\lt Y\leq d)$
    Since $(X=a,-\infty\lt Y\leq c)$ and $(X=a,c\lt Y\leq d)$ are mutually exclusive then
    $P(X=a,-\infty\lt Y\leq d) =P(X=a,-\infty\lt Y\leq c)+P(X=a,c\lt Y\leq d)$
    $\implies F(a,d)-F(a-0,d) =F(a,c)-F(a-0,c)+P(X=a,c\lt Y\leq d)$
    $\implies P(X=a,c\lt Y\leq d)=F(a,d)-F(a-0,d) -F(a,c)+F(a-0,c)$
    $\implies P(X=a,c\lt Y\leq d)=F(a,d)+F(a-0,c)-F(a-0,d) -F(a,c)$
    

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12. $P(a\lt Y\leq b, Y=c)=F(b,c)+F(a,c-0)-F(b,c-0) -F(a,c)$ where $a\lt b$
    Let $c\lt d$. We have $F(x,c)-F(x,c-0)=P(-\infty\lt X\leq x,Y=c)$
    Also $(-\infty\lt X\leq b,Y=c) =(-\infty\lt X\leq a,Y=c)+(a\lt X\leq b,Y=c)$
    Since $(-\infty\lt X\leq a,Y=c)$ and $(a\lt X\leq b,Y=c)$ are mutually exclusive then
    $P(-\infty\lt X\leq b,Y=c) =P(-\infty\lt X\leq a,Y=c)+P(a\lt X\leq b,Y=c)$
    $\implies F(b,c)-F(b,c-0) =F(a,c)-F(a,c-0)+P(a\lt X\leq b,Y=c)$
    $\implies P(a\lt X\leq b,Y=c)=F(b,c)-F(b,c-0) -F(a,c)+F(a,c-0)$
    $\implies P(a\lt X\leq b,Y=c)=F(b,c)+F(a,c-0)-F(b,c-0) -F(a,c)$
    

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13. $P(X=a, Y=c)=F(a,c)+F(a-0,c-0)-F(a,c-0) – F(a-0,c)$
    We have $P(-\infty\lt X\leq x,Y=c)=F(x,c)-F(x,c-0)$
    Now $(-\infty\lt X\leq a,Y=c)= (-\infty\lt X\lt a,Y=c)+(X=a, Y=c)$
    Since $(-\infty\lt X\lt a,Y=c)$ and $(X=a, Y=c)$ re mutually exclusive then
    $P(-\infty\lt X\leq a,Y=c)= P(-\infty\lt X\lt a,Y=c)+P(X=a, Y=c)$
    $\implies F(a,c)-F(a,c-0) = F(a-0,c)-F(a-0,c-0)+P(X=a, Y=c)$
    $\implies P(X=a, Y=c)= F(a,c)-F(a,c-0) – F(a-0,c)+F(a-0,c-0)$
    $\implies P(X=a, Y=c)= F(a,c)+F(a-0,c-0)-F(a,c-0) – F(a-0,c)$
    

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Applications #

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  Two Dimensional Distribution Functions are used in econometrics, physics, and artificial intelligence. They are widely applied in modeling dependencies between financial assets, predicting correlated events in data science, and analyzing joint distributions in reliability engineering. Their significance in probability and statistics makes them a valuable analytical tool.

Conclusion #

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  Understanding Two Dimensional Distribution Functions is crucial for statistical modeling and data analysis. These functions provide insights into the joint behavior of random variables, enabling efficient probability computations. Their applications in multiple fields highlight their importance in probability and statistical research.

Frequently Asked Questions #

1. What are Two Dimensional Distribution Functions?
   Two Dimensional Distribution Functions describe the probability distribution of two random variables and their joint behavior.
2. Why are Two Dimensional Distribution Functions important?
   They are important because they help in understanding dependencies between two variables and are used in statistical modeling.
3. How are Two Dimensional Distribution Functions applied in finance?
   These functions are applied in finance to model dependencies between assets and to assess portfolio risk.
4. What is the difference between marginal and joint distribution?
   A joint distribution considers two variables together, while a marginal distribution focuses on a single variable independently.
5. Can Two Dimensional Distribution Functions be used in AI?
   Yes, they are used in AI to model relationships between correlated features in machine learning models.
6. What is the role of covariance in Two Dimensional Distribution Functions?
   Covariance measures the degree to which two variables change together and is an essential component of joint distributions.
7. How do Two Dimensional Distribution Functions help in econometrics?
   They are used to model economic relationships and to predict trends based on statistical dependencies.
8. What are the common types of Two Dimensional Distribution Functions?
   Common types include joint probability mass functions, joint probability density functions, and cumulative distribution functions.
9. Do Two Dimensional Distribution Functions affect correlation?
   Yes, they provide insight into correlation coefficients and the strength of relationships between two variables.
10. Where can Two Dimensional Distribution Functions be studied in detail?
    They can be studied in probability textbooks, online statistical courses, and research papers focused on statistical analysis.