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