One notable feature of the Poisson distribution, as distinct from the Gaussian, is that it has no left tail whatsoever, as the distribution goes to 0 (and stays there) at 0 or below.
And it’s not necessarily true that the mean of a Poisson is equal to the variance. It can be, but it can also be much larger or smaller. A Poisson distribution with a mean much greater than the standard deviation ends up looking very much like a Gaussian, but one with a small mean (relative to the standard deviation) looks very different.
That is very different from what I learned, would appreciate a cite on that.
One other nice feature of Poisson regressions is that they are additive, meaning if you fit multiple (independent) regressions you can use the added means to create a combined distribution.
The mean and variance of a Poisson distribution are always equal essentially by definition. That is the distribution is defined by its probability mass and the mean and variance are equal to the parameter usually called lambda.
But they you switch and use mean and standard deviation. When the mean is much larger than that standard deviation (not variance), then lambda is large and the skewness is 1/sqrt(lambda) and the kurtic excess is 1/lambda. In this sense the distribution for large lambda is more Gaussian than for small lambda.
I switched to standard deviation because I assumed that “variance” was a mistake, because mean and variance can’t be the same, as they have different dimension.
This is incorrect. (wow the thrill of being able to correct the great Chronos. I feel unworthy)
I understand where your coming from, in the case of scaled parameters you would need your standard deviation to be on the same scale as your mean. So if you have a life time of 9 months your standard deviation might be 3 months and so your variance would have to be 9 square months. If you changed the units to years, they your lifetime would be 0.75 years, your standard deviation would be 0.25 years and your variance would be 0.0625 squared years. So the variance that was equal to the mean in one set of units couldn’t be equal in an alternate set.
In the case of the poisson distribution you are dealing with count data so there isn’t any issue with needing units to match. The Poisson distribution only has one parameter: the mean. All the rest of the characteristics of the distribution fall from that.
If your geiger counter is getting an average of 10 hits per second, then the variance of the number counts after one second will be 10. This comes to an average of 600 counts per minute which will have a variance also of 600.
This might seem counter intuitive since if the mean increase by a factor of 60, shouldn’t the variance increase by a factor of 60^2. But the extra time gives more time for the noise to cancel out and so the relative signal to noise ratio will be smaller. after a minute than it would be after a second.
They are ‘normal’ in that context but the term has come to be used in statistics specifically to refer to the (Laplace-)Gauss distribution, presumably because it is the most frequently encountered in introductory statistics courses and widely used in frequentist statistics such as hypothesis testing, error analysis, and basic reliability modeling.
I imagine that, functionally, that term is used for that distribution because by this point, it’s the name for that distribution. The rest is etymology, more than mathematics, and I suspect that most folks who use the term don’t know the etymology.
It is etymologically coincidental that the standard Gaussian distribution is both mathematically normal (perpendicular) and is the ‘typical’ distribution that people re familiar with (in many cases of casual user of statistics, the Gaussian distribution may be the only distribution they are familiar with). It is like how when you use the term “trigonometry” people think in terms of sine, cosine, and tangent functions and their inverses because those are the most commonly used in a basic analytic geometry class or in studying Fourier analyssi but there are actually three other much less used trigonometric functions (secant, cosecant, and cotangent) that are also included in the domain of trigonometry.
I have heard of a definition of skewness which is basically the third standardized moment about the mean. E.g. for any distribution which is symmetric with respect to the mean, all the odd central moments trivially vanish, so the skewness of such a distribution will be zero.
As for a “normally distributed” variable, I am pretty sure people are justifiably going to assume the distribution is Gaussian.
But this is true of any symmetric distribution for which the variance exists so the Gaussian distribution is only a normal distribution rather than the normal distribution by this logic.
