Jun 8, 2021 - 14:34
An infinite series problem.
By Vamshi Jandhyala in mathematics
June 8, 2021
Problem
— 級数bot (@infseriesbot) June 8, 2021
Solution
Let $X$ be a Poisson random variable with parameter $n$. The given expression is equal to $\mathbb{P}[X \leq n]$. For large $n$, it is well known that $Poisson(n) \approx Normal(n,n)$. Therefore the limit of the expression is $\frac{1}{2}$ as the probability of a Normal variable being less than or equal to its mean is $\frac{1}{2}$.