Q:

Concentration of poisson distribution

Let $X$ be a Poisson random variable with mean $\lambda$. How do I show that $P[X \geq \lambda] = 1/2$? Also, I was wondering what distributions have this property that the density is concentrated equally around the mean. I see that it's true for Gaussian and Uniform distributions.

A:

P[X ≥ μ] = 1/2 if and only if the mean (μ) and median "coincide". This is true for Gaussian and uniform distribution, but not always true for Poisson.
There is exactly one μ between every unit interval [n, n+1) such that P[X ≥ μ] = 1/2. Some of them are: ln 2, 1.6784…, 2.674…, 3.672…, ...

