Super short Uncertainty Wednesday post today as I have a crazy busy week featuring our annual meeting (where we meet with our Limited Partners). The last few weeks we have been digging into sample means and expected values. We saw some surprising things already, such as random variables that do not have an expected value.
During these posts I have sometimes used the terms random variable and probability distribution interchangeably, despite previously having given two separate definitions (see links). So what gives? Technically they are different concepts but for some commonly used probability distributions, such as the normal distribution, all random variables based on the distribution differ only in one or two parameters (for the normal distribution: mean and standard deviation).
These differences turn out to be boring and so using the terms interchangeably seems OK. Put differently, often the difference between distributions is more important than the difference between the same distribution with different parameters. For instance, all random variables based on Cauchy distributions (fat tailed) are very different from all random variables based on Normal distributions. That difference is huge compared to the difference between normally distributed random variables.
Important caveat: there are distributions for which changes in the parameter make a big difference, such as the power law distribution. In those cases random variables based on the same distribution will differ a lot from each other.
