Overview
Stochastic demand modeling begins by assuming that demand is a random variable, and then proceeds to assign and fit a distibution to that variable based on past history.
Distribution
Once you have decided to model demand as a stochastic variable, you need to assign it a distribution. A common distribution to use is the normal distribution. The normal distribution is a standard distribution for many random variables, primarily because of the central limit theorem.
In the case of product demand, it is a reasonable assumption, but not perfect. The normal distribution will assign positive probabilities to negative demand, and will assign postive (although very small) probability to demand that is arbitrarily large.
If the unbounded nature of the normal distribution is unacceptable, the lognormal distribution, which is bounded below, or the beta distribution, which is bounded both above and below are sometimes used as alternatives.
Fitting the Distribution
Once you assume a distribution function for demand, you will need to fit the distribution to data. This typically involves fitting a set of parameters that are used to specify the distribution, such as the mean and variance.
The easiest way to accomplish this is to take the mean and variance from a dataset representing real customer demand. (see the method of moments)
Note, the demand being modeled by necessity is the customer demand over a given time frame. When fitting the data, one must make certain that the demand in the dataset represents a consistent time frame. If one is using a normal distribution to model demand, it is a simple matter to model demand over longer periods using the sum of normal variables formula.
Demand, Price and Time
The foregoing analysis assumes that the price is fixed, and the time frame is fixed. That is, the distribution is really a conditional distribution.
{% D \sim f(q | t, p) %}
The typical way to extend the analysis to include price and time is to assume that the
expected value
of demand is a function of price and time.
{% \mathbb{E}(q|t, p) = f(t,p) %}
Changes in price are assumed to create a shift in the expectation based on some estimate of
price elatsticity.
Time dependence utiliizes the tools of time series analysis and typically models the time dependence of demand as a trend and seasonality.