From the Poisson Distribution to Stirling's Approximation

The Poisson distribution is the most famous probability model for counts , non-negative integer values. Many real-world phenomena are well approximated by this distribution, including the number of German bombs that landed in 1/4km grid squares in south London during WWII. Formally, we say that a discrete random variable \(X\) follows a Poisson distribution with rate parameter \(\mu > 0\) , abbreviated \(X \sim \text{Poisson}(\mu)\) , if \(X\) has support set \(\{0, 1, 2, ...\}\) and probability