Probability Distribution: Binomial, Poisson and Normal+

Application of Distributions in Probability

Binomial Distribution

Definition:

• Conditions: Fixed number of trials, independent outcomes, two possible outcomes.
• Formula: P(X=k) = $\left(\begin{array}{c}n\\ k\end{array}\right).p^k.(1-p{)}^{n-k}$
• Example Application:
  • Laboratory Performance: Evaluating quality in chemical analyses of environmental samples.
  • Use: Assessing if reported values deviate significantly from expected values, using standard deviations.

Poisson Distribution

Definition:

• Conditions: Occurrence of events in a fixed interval of time or space, with events occurring independently at a constant rate.
• Formula: P(X=k) = $\frac{{\lambda }^k{e}^{-\lambda }}{k!}$
• Example Application:
  • Rare Events: Modeling the number of arrivals at a service point, such as customers arriving at a bank per hour.

Normal Distribution

Definition:

• Conditions: Continuous probability distribution describing data that cluster around a mean value, forming a symmetric bell-shaped curve.
• Formula: $f(x\mid \mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi \sigma {}^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$
• Example Application:
  • Statistical Analysis: Used in quality control to monitor production processes, ensuring products meet specified standards.

Additional Information:

• Statistical Tools: These distributions are crucial in various fields for modeling and analyzing data.
• Real-World Applications: From quality assurance in laboratories (binomial) to modeling rare occurrences (Poisson) and analyzing continuous data (normal), distributions provide insights into probability and uncertainty.

Each distribution serves distinct purposes based on the nature of the data and the specific questions being addressed, making them indispensable tools in statistical analysis and decision-making processes.