I am currently working on some error analysis homework and I am having trouble understanding some basic concepts. In particular I don't understand if there is any difference between the sample mean, standard deviation, standard error if I have a poisson distribution as opposed to a normal distribution. Are they calculated the same way? Intuitively I would say that there is no difference but I am not really sure. Maybe someone can enlighten me.
Thanks in advance.
Best Answer
For any distribution with unknown mean $\mu$ and variance $\sigma^2$,
and this is also true for a sample Poisson distribution. So you can build your sample mean, sample standard deviation and estimate of the standard error of the mean this way if you wish.
But a property of Poisson distributions is that the population mean and variance are in fact equal, so if your aim is estimating that parameter, then producing these two numbers might not be so helpful. Since the sample mean is a sufficient statistic for the Poisson distribution, you might prefer to use $\bar{X}$ as an estimator of the variance, $\sqrt{\bar{X}}$ as an estimator of the standard deviation, and $\sqrt{\frac{\bar{X}}{n}}$ for the standard error of the mean.
If your ultimate aim is constructing confidence intervals Comparison of Confidence Intervals for the Poisson Mean: Some New Aspects., Patil, V. V. & Kulkarni, H. V. has $19$ methods