Why is the poisson distribution more symmetrical with a higher mean?

Question

So say we have two data points, one is $x=0, y=100$ and the other is $x=1,y= 0$, thus the mean would be $50$. Now obviously, the distribution would not look very symmetrical. But why would a higher mean result in a more symmetrical distribution (aka. normal distribution)? Imagine again two points $x=0, y=1000$ and $x=1,y=0$, the mean would now be $500$ (higher) yet the distribution would look even less symmetrical..? Why is it then said that a higher mean = more symmetrical distribution?

Am I missing something or did I understand this wrongly?

Thanks in advance for any help!

Answer 1

• The Poisson distribution takes values on $0,1,2,3,\ldots$. No matter what the mean is, it isn't symmetric because to the right of the mean, it has a tail tending to infinity, while to the left it gets "blocked" by the "wall" at zero. However, you are probably asking why it appears symmetric for large values of the mean.
• The Poisson distribution has standard deviation equal to the square root of the mean. So when the mean is large, most of the probability mass is concentrated on the interval $[\mu - 2\sqrt{\mu}, \mu + 2\sqrt{\mu}]$ (roughly speaking), which is far from zero. Outside of this interval, the probabilities are quite small, so you cannot see the asymmetry between the "wall" at zero and the long tail to infinity. On the other hand for small values of $\mu$, the wall $0$ is within one or two standard deviations of the mean, so the "hump" of the distribution is pressed against the "wall" at zero, making the distribution look asymmetric.