Prediction problem by learning a distribution If I learn a distribution of random variables from a sequence of data, how can I use the distribution to forecast the data? The current data in sequence may or may not have any relationship with previous data.
 A: If you have the data then you can compute a histogram approximating the probability density function (pdf) also known as a frequency distribution . Drawing random numbers from this pdf will not generate useful data unless the observations are assumed to have a temporal independence (among other characteristics). The pdf distribution is insufficient as one often (always !) needs to have the acf of the values in order to correctly forecast the data. This is why we use the actual observed data to form two descriptive statistics the pdf and the acf. The acf can be used to identify the form/nature of the auto-projective structure while taking into account pulses/level shifts/local time trends/seasonal pulses/changes in parameters over time/changes in error variance over time.
Now given a useful model (arima plus ...) one can then forecast.
The idea of using the observed histogram/frequency distribution/pdf alone is only correct if the series is independent (random process) from observation to observation
