Exact Binomial Probability Mass Function Simplified

Background:The binomial distribution is fundamental in probability theory and statistics. While approximations such as the Poisson distribution are useful, an exact algebraic simplification can be advantageous. Material and methods:For large sample sizes \(n\), computational precision issues may arise even when using the exact binomial formula. Results:An alternative exact representation of the binomial PMF using the substitution \(x = n - k\) and Pochhammer symbol or the rising factorial (x), leading to a form that separates the factorial ratio into a product and highlights the role of the complement probability is derived. The resulting expression is\begin{align}\boxed{P(X = k) = \frac{(k+1)^{(x)}}{x!} \cdot (1-p)^{x} \cdot e^{k \ln p}}\end{align}Conclusion:The simplified exact formulation of the binomial probability mass function offer various computational and other benefits in certain settings.