Solved – How is Poisson distribution different to normal distribution

distributionshistogramnormal distributionpoisson distribution

I have generated a vector which has a Poisson distribution, as follows:

x = rpois(1000,10)

If I make a histogram using hist(x), the distribution looks like a the familiar bell-shaped normal distribution. However, a the Kolmogorov-Smirnoff test using ks.test(x, 'pnorm',10,3) says the distribution is significantly different to a normal distribution, due to very small p value.

So my question is: how does the Poisson distribution differ from a normal distribution, when the histogram looks so similar to a normal distribution?

Best Answer

  1. A Poisson distribution is discrete while a normal distribution is continuous, and a Poisson random variable is always >= 0. Thus, a Kolgomorov-Smirnov test will often be able to tell the difference.

  2. When the mean of a Poisson distribution is large, it becomes similar to a normal distribution. However, rpois(1000, 10) doesn't even look that similar to a normal distribution (it stops short at 0 and the right tail is too long).

  3. Why are you comparing it to ks.test(..., 'pnorm', 10, 3) rather than ks.test(..., 'pnorm', 10, sqrt(10))? The difference between 3 and $\sqrt{10}$ is small but will itself make a difference when comparing distributions. Even if the distribution truly were normal you would end up with an anti-conservative p-value distribution:

    set.seed(1)

hist(replicate(10000, ks.test(rnorm(1000, 10, sqrt(10)), 'pnorm', 10, 3)$p.value))

enter image description here

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