An example of applications of the Poisson distribution could be the counting of the number of \( \alpha \)-particles emitted from a radioactive source in a given time interval. In the limit of \( n\rightarrow \infty \) and for small probabilities \( y \), the binomial distribution approaches the Poisson distribution. Setting \( \lambda = ny \), with \( y \) the probability for an event in the binomial distribution we can show that $$ \begin{equation*} \lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. \end{equation*} $$