I have a poisson random Variable $X$ with parameter $\lambda$. I need to find the value of $P(X \leq \lambda)$.
I know that, this is given by
$$P(X \leq \lambda) = e^{-\lambda} \sum_{x=0}^{\lambda}{\frac{\lambda^x}{x!}}$$
How to calculate further? I found an answer which says this value is 0.5, but it calculates the value numerically using a R script. Is it possible to calculate this value directly? Thanks!
That formula is valid when $\lambda$ is an integer. In general $$P(X\le\lambda)=e^{-\lambda}\sum_{n=0}^{\lfloor\lambda\rfloor}{\frac{\lambda^n}{n!}}$$ where $\lfloor\lambda\rfloor$ is the integer part of $\lambda$. This will not equal $\frac12$ in general, but for large $\lambda$ will be approximately $\frac12$.