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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.

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    Not sure if $P(X \geq \lambda) = 1/2$. Take $\lambda = 1$. Then $P(X \geq 1) = 1-e^{-1}$.2010-10-24

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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…, ...

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    @Naga: Obviously, that proof is not correct for small $n$. The larger $n$ is, though, the closer P(N>n) gets to 1/2. Are you maybe only interested in asymptotics?2010-10-26