Distribution Functions have been extensively used in probability theory and statistics. Their importance in defining probability distributions has been acknowledged for centuries. Various fields, including economics, engineering, and data science, have benefited from their applications. The study of Distribution Functions has provided a foundation for statistical modeling and risk assessment.
Introduction #
Distribution Functions play a crucial role in probability and statistics. These functions define how probabilities are distributed over different values of a Random Variable. Both cumulative and probability density functions are used to describe probability distributions. Many real-world problems are analyzed using Distribution Functions.
Distribution Function #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\). The Distribution Function of a random variable \(X\) is a function , denoted by \(F(x) \) or \(F_{x}(x) \) defined by \(F:R\to [0,1] \) such that \(F(x)=P(-\infty\lt X \leq x) \).
Properties #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\). Let \(F(x) \) be the distribution function of a random variable \(X\).
- If \(a\lt b\) then \(P(a\lt X\leq b)=F(b)-F(a) \).
Let \(a\lt b\).
We have \((-\infty\lt X\leq a)+(a\lt X\leq b)= (-\infty\lt X\leq b) \)
Since \( (-\infty\lt X\leq a)\) and \( (a\lt X\leq b)\) are mutually exclusive events then
\(P((-\infty\lt X\leq a)+(a\lt X\leq b))=P(-\infty\lt X\leq b) \)
\(\implies P(-\infty\lt X\leq a)+P(a\lt X\leq b)=P(-\infty\lt X\leq b) \)
\(\implies F(a)+P(a\lt X\leq b)=F(b) \)
\(\implies P(a\lt X\leq b)=F(b)-F(a) \)
- The distribution function \(F(x) \) is monotone non-decreasing.
Let \(a\lt b\).
Then we have \(F(b)-F(a)=P(a\lt X\leq b)\).
\(\implies F(b)-F(a)\geq 0 \because P(a\lt X\leq b)\geq 0\).
\(\implies F(b)\geq F(a) \) when \(a\lt b\)
\(\therefore \) \(F(x) \) is monotone non-decreasing.
- \(F(-\infty)=0 \)
Let \(A_{n}\) be the event \( (-\infty\lt X\leq -n) \), \(n=1,2,… \).
Then \(A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…\).
Therefore \(\lim\limits_{n\to \infty }A_{n}=O \)
\(\implies P \big(\lim\limits_{n\to \infty }A_{n} \big)=P(O)=0 \)
\(\implies \lim\limits_{n\to \infty }P(A_{n})=0 \)
\(\implies \lim\limits_{n\to \infty }P(-\infty\lt X\leq -n)=0 \)
\(\implies \lim\limits_{n\to \infty }F(-n)=0 \)
\(\implies F(-\infty)=0 \)
- \(F(\infty)=1 \)
Let \(A_{n}\) be the event \( (-\infty\lt X\leq n) \), \(n=1,2,… \).
Then \(A_{1}\subseteq A_{2}\subseteq … \subseteq A_{n}\subseteq…\).
Therefore \(\lim\limits_{n\to \infty }A_{n}=(-\infty\lt X\lt \infty)=S \).
\(\implies P \big(\lim\limits_{n\to \infty }A_{n} \big)=P(S)=1 \)
\(\implies \lim\limits_{n\to \infty }P(A_{n})=1 \)
\(\implies \lim\limits_{n\to \infty }P(-\infty\lt X\leq n)=1 \)
\(\implies \lim\limits_{n\to \infty }F(n)=1 \)
\(\implies F(\infty)=1 \)
- \(F(a)-F(a-0)=P(X=a) \)
Let \(A_{n}\) be the event \( (a-\frac{1}{n}\lt X\leq a) \), \(n=1,2,… \).
Then \(A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…\).
Therefore \(\lim\limits_{n\to \infty }A_{n}=(X=a) \)
\(\implies P \big(\lim\limits_{n\to \infty }A_{n} \big)=P(X=a) \)
\(\implies \lim\limits_{n\to \infty }P(A_{n})=P(X=a) \)
\(\implies \lim\limits_{n\to \infty }P(a-\frac{1}{n}\lt X\leq a)=P(X=a) \)
\(\implies \lim\limits_{n\to \infty } \Big[F(a)-F(a-\frac{1}{n})\Big]=P(X=a) \)
\(\implies F(a)-F(a-0)=P(X=a)\)
- \( F(a+0)=F(a) \)
Let \(A_{n}\) be the event \( (a\lt X\leq a+\frac{1}{n}) \), \(n=1,2,… \).
Then \(A_{1}\supseteq A_{2}\supseteq … \supseteq A_{n}\supseteq…\).
Therefore \(\lim\limits_{n\to \infty }A_{n}=O \)
\(\implies P \big(\lim\limits_{n\to \infty }A_{n} \big)=P(O) \)
\(\implies \lim\limits_{n\to \infty }P(A_{n})=0 \)
\(\implies \lim\limits_{n\to \infty }P(a\lt X\leq a+\frac{1}{n})=0 \)
\(\implies \lim\limits_{n\to \infty } \Big[F(a+\frac{1}{n})-F(a)\Big]=0 \)
\(\implies F(a+0)-F(a)=0\)
\(\implies F(a+0)=F(a)\)
- If \(a\lt b\) then \(P(a\lt X\lt b)=F(b-0)-F(a) \).
Let \(a\lt b\).
We have \( P(a\lt X\leq b)=F(b)-F(a) \)
Now \((a\lt X\leq b)=(a\lt X\lt b)+(X=b) \)
Since \((a\lt X\lt b) \) and \((X=b) \) are mutually exclusive events then
\(P(a\lt X\leq b)=P(a\lt X\lt b)+P(X=b) \)
\( \implies F(b)-F(a)=P(a\lt X\lt b)+F(b)-F(b-0) \)
\( \implies P(a\lt X\lt b) =F(b-0)-F(a) \)
- If \(a\lt b\) then \(P(a\leq X\lt b)=F(b-0)-F(a-0)\).
Let \(a\lt b\).
We have \( P(a\lt X\lt b)=F(b-0)-F(a) \)
Now \((a\leq X\lt b)=(a\lt X\lt b)+(X=a) \)
Since \((a\lt X\lt b) \) and \((X=a) \) are mutually exclusive events then
\(P(a\leq X\lt b)=P(a\lt X\lt b)+P(X=a) \)
\( \implies P(a\leq X\lt b)=F(b-0)-F(a)+F(a)-F(a-0) \)
\( \implies P(a\leq X\lt b) =F(b-0)-F(a-0) \)
- \(P(X\lt a)=F(a-0) \).
Now \((X\lt a)=(-\infty\lt X\lt a) \)
and \( (-\infty\lt X\leq a)=(-\infty\lt X\lt a)+(X=a) \)
Since \((-\infty\lt X\lt a) \) and \( (X=a)\) are mutually exclusive events then
\( \implies P(-\infty\lt X\leq a)=P(-\infty\lt X\lt a)+P(X=a) \)
\( \implies F(a) =P(-\infty\lt X\lt a)+F(a)-F(a-0) \)
\( \implies P(-\infty\lt X\lt a)=F(a-0) \)
\( \implies P( X\lt a)=F(a-0) \)
- \(P(X\gt a)=1-F(a-0) \).
We have \( P( X\lt a)=F(a-0) \)
Now \(P(X\gt a)=1-P(X\lt a) \)
\( \implies P(X\gt a)=1-F(a-0) \)
Applications #
Distribution Functions are widely applied in finance, engineering, and medical research. In finance, they are used to model stock market fluctuations. In engineering, reliability assessments of materials and systems rely on probability distributions. Additionally, predictive modeling in machine learning is based on probability distributions.
Conclusion #
Distribution Functions serve as a fundamental concept in probability and statistics. Their applications in different domains help in making informed decisions and analyzing uncertainties. Understanding Distribution Functions allows better statistical modeling and predictive analysis. The significance of these functions in real-world applications continues to grow.
FAQs #
- What are Distribution Functions?
Distribution Functions describe how probabilities are assigned to different values of a Random Variable. - Why are Distribution Functions important?
They help in understanding probability distributions and making statistical inferences. - What are the types of Distribution Functions?
The main types include cumulative distribution functions (CDF) and probability density functions (PDF). - How are Distribution Functions used in finance?
They are used to model stock price movements and risk assessments. - How do engineers use Distribution Functions?
Engineering applications include reliability testing and failure rate analysis. - What is the role of Distribution Functions in machine learning?
Machine learning algorithms use probability distributions to make predictions. - Can Distribution Functions be used in medical research?
Yes, they are used to model disease spread and patient outcomes. - How is a cumulative Distribution Function different from a probability density function?
A CDF gives the probability that a Random Variable takes a value less than or equal to a specific value, while a PDF shows probability density. - What is an example of a commonly used Distribution Function?
The normal distribution is one of the most commonly used Distribution Functions. - How are Distribution Functions related to Random Variables?
They define how probabilities are assigned to possible values of Random Variables.
