Discrete Distributions have been studied extensively in probability and statistics. Their significance in modeling countable outcomes has been widely recognized. Various fields, including actuarial science, computer science, and economics, have utilized these distributions. The study of Discrete Distributions has enhanced statistical modeling and decision-making processes.
Introduction #
Discrete Distributions describe probability distributions for countable outcomes. These distributions, including the binomial, Poisson, and geometric distributions, are widely applied in statistical modeling. Many real-world phenomena, such as defect rates in manufacturing and customer arrivals at a service center, are analyzed using Discrete Distributions.
Discrete Distribution #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\) and \(X\) be a random variable. The probability distribution of \(X\) is said to be Discrete if the spectrum of \(X\) takes a discrete values \(…,x_{-2},x_{-1},x_{0},x_{1},x_{2},… \).
Properties #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\) and \(X\) be a random variable. Let the spectrum of \(X\) takes a discrete values \(…,x_{-2},x_{-1},x_{0},x_{1},x_{2},… \) with \(P(X=x_{i})=f_{i} \)
- If \( x_{i}\leq x \lt x_{i+1}\) then \(F(x)=\displaystyle\sum_{j=-\infty}^{i} f_{j} \)
Let \( x_{i}\leq x \lt x_{i+1}\).
We have \(F(x)=P(-\infty\lt X \leq x) \)
\(\implies F(x)=P(…+(X=x_{i-1})+(X=x_{i})) \)
\(\implies F(x)=P\Big(\displaystyle\sum_{j=-\infty}^{i} (X=x_{j}) \Big) \)
\(\implies F(x)=\displaystyle\sum_{j=-\infty}^{i} P(X=x_{j}) \)
\(\implies F(x)=\displaystyle\sum_{j=-\infty}^{i} f_{j} \)
- \(\displaystyle\sum_{j=-\infty}^{\infty} f_{j}=1 \)
We have \(F(\infty)=1 \)
\(\implies F(-\infty\lt X \lt \infty)=1 \)
\(\implies P(…+(X=x_{i-1})+(X=x_{i})+…)=1 \)
\(\implies P\Big(\displaystyle\sum_{j=-\infty}^{\infty} (X=x_{j}) \Big)=1 \)
\(\implies \displaystyle\sum_{j=-\infty}^{\infty} P(X=x_{j}) =1 \)
\(\implies \displaystyle\sum_{j=-\infty}^{\infty} f_{j} =1 \)
- \(P(X=a)=0 \) for any non spectrum point \(a\)
Let \(a\) be any non spectrum point.
Then \( \exists \) two step points \( x_{i} \) and \( x_{i+1} \) such that \( x_{i} \lt a \lt x_{i+1} \).
Since \( x_{i} \lt a-0 \lt a \lt x_{i+1} \)
then for \( x_{i} \lt a-0 \lt x_{i+1} \), \(F(a-0)= \displaystyle\sum_{j=-\infty}^{i} f_{j} \).
Also for \( x_{i} \lt a \lt x_{i+1} \), \(F(a)= \displaystyle\sum_{j=-\infty}^{i} f_{j} \).
\(\implies F(a)=F(a-0)=\displaystyle\sum_{j=-\infty}^{i} f_{j} \)
\(\implies F(a)-F(a-0)=0\)
\(\implies P(X=a)=0 \) - \(P(a\lt X \leq b)=\displaystyle\sum_{a\lt x_{j} \leq b }^{} f_{j} \)
Let \(a\lt b\).
Then \( \exists \) two distinct step points \( x_{k} \) and \( x_{t} \) such that \( x_{k} \leq a \lt b \lt x_{t} \).
\(\implies x_{k} \leq a \lt x_{k+1} \leq x_{t-1} \leq b \lt x_{t} \).
We have \(F(x)=\displaystyle\sum_{j=-\infty}^{i} f_{j} \) where \(x_{i}\leq x \lt x_{i+1} \)
Now \(P(a\lt X \leq b)=F(b)-F(a) \)
\(\implies P(a\lt X \leq b)=\displaystyle\sum_{j=-\infty}^{t-1} f_{j}-\displaystyle\sum_{j=-\infty}^{k} f_{j} \)
\(\implies P(a\lt X \leq b)=\displaystyle\sum_{j=k+1}^{t-1} f_{j}\)
\(\implies P(a\lt X \leq b)=\displaystyle\sum_{x_{j}=x_{k+1}}^{x_{t-1}} f_{j}\)
\(\implies P(a\lt X \leq b)=\displaystyle\sum_{a\lt x_{j} \leq b }^{} f_{j}\)
Binomial \((n,p) \) Distribution #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\) and \(X\) be a random variable. Let the spectrum of \(X\) takes a discrete values \(0,1,2,…,n \) with \(P(X=i)=f_{i} \), \((i=0,1,2…,n) \). The random variable \(X\) is said to binomial \((n,p) \) distribution if \( f_{i}=\tbinom{n}{i}p^{i}(1-p)^{n-i}\) where \(n\) is a positive integer and \( 0\lt p \lt 1\).
Possion \(\mu \) Distribution #
Let \(E\) be a random experiment and \(S\) be the event space of \(E\) and \(X\) be a random variable. Let the spectrum of \(X\) takes a discrete values \(0,1,2,… \) with \(P(X=i)=f_{i} \), \((i=0,1,2…) \). The random variable \(X\) is said to possion \(\mu \) distribution if \( f_{i}= \frac{\mu^{i}}{i!}e^{-\mu} \) where \( \mu \) is a positive integer.
Applications #
Discrete Distributions are widely applied in fields such as finance, engineering, and healthcare. In finance, they are used to model default probabilities. In engineering, system reliability assessments rely on discrete probability models. Additionally, disease prediction models in healthcare use Discrete Distributions to analyze occurrences.
Conclusion #
Discrete Distributions play a fundamental role in probability and statistics. Their applications help in risk analysis, quality control, and predictive modeling. A deeper understanding of these distributions allows better decision-making in uncertain scenarios. Their importance in data-driven analysis continues to expand.
FAQs #
- What are Discrete Distributions?
Discrete Distributions describe probability distributions where outcomes are countable. - Why are Discrete Distributions important?
They help in modeling real-world scenarios involving countable data points. - What are common types of Discrete Distributions?
The binomial, Poisson, and geometric distributions are commonly used Discrete Distributions. - How are Discrete Distributions used in finance?
They are used to model credit risks and investment returns. - How do engineers use Discrete Distributions?
They are applied in quality control and system reliability analysis. - What is the role of Discrete Distributions in healthcare?
They are used in epidemiological studies to predict disease outbreaks. - Can Discrete Distributions be applied to machine learning?
Yes, they are used in classification algorithms and probabilistic modeling. - How is a binomial distribution different from a Poisson distribution?
A binomial distribution models a fixed number of trials, while a Poisson distribution models rare events over time. - What is an example of a commonly used Discrete Distribution?
The Poisson distribution is frequently used to model event occurrences. - How are Discrete Distributions related to probability theory?
They define how probabilities are assigned to specific countable outcomes.
