There’s two fundamental steps to selecting the very best distribution of the input variable. The very first is to sort out how good you really have to define the distribution. The second reason is to define a distribution with very little bias as you possibly can.
Not every inputs have to be fully defined like entropy do. If how they communicate with one another has certain characteristics, then it’s only the mean and also the standard deviation that should be known.
The important thing for this step is a vital limit theorem. When the product is one the adds or multiplies the inputs, then your central limit theorem states the output is a Normal or Lognormal distribution correspondingly. These two distributions are defined only by two parameters. Which means that just the mean and also the standard deviation are necessary to define all of them.
Furthermore, the processes of adding and multiplying only depend upon the mean and also the standard deviation. When the inputs are added or multiplied, then just the mean and standard deviation of every input is required to calculate the mean and also the standard deviation from the output.
Therefore, since the results of multiplications and additions is placed (Lognormal or Normal correspondingly) and just the mean and standard deviation are in play, just the mean and standard deviation have to be noted for input variables which are either exclusively multiplied or added together. Observe that this assumes the needs from the central limit theorem are met.
When merely a limited quantity of details are available, you will find limits towards the distributions that may be selected.
For instance, if perhaps the utmost and minimum possible values are known, a uniform distribution ought to be selected. This is actually the least biased distribution you can use within this situation. It’s a distribution that maximises the uncertainty given what’s known. This really is known as an optimum entropy distribution which is the kind which should if bias will be removed.
For an additional example, if perhaps the mean and standard deviation are known then your least biased (maximum entropy) distribution may be the Normal distribution.
The greater information which is had, the greater accurate the defined distribution could be. The easiest method to make sure that an impartial distribution can be used would be to collect just as much data as you possibly can, calculate the moments that may be reasonably calculated after which discover the maximum entropy distribution.