# Random variable

Using the martingale method of forecast evolution (MMFE) of heath and Jackson and obtain a formula for the mean and variance of the lead time demand and apply these formulas to specific demand models.

The Martingale Method of Forecast Evolutions (MMFE) can be represented as follows:

(SEE ATTACHMENT)

Using the martingale method of forecast evolution (MMFE) of heath and Jackson and obtain a formula for the mean and variance of the lead time demand and apply these formulas to specific demand models.

The Martingale Method of Forecast Evolutions (MMFE) can be represented as follows:

At the beginning of the first period, there is an initial forecast for the demand to prevail

in period s for each s >= 1 in the planning horizon. Forecasts are updated at the beginning

of every period as follows: For all s >= t

+

where is the forecast at the beginning of period t for the demand to prevail during

period s >= t and is a mean zero, variance ,random variable that becomes known

at the end of period t. The actual demand seen during period s is Ds = .Thus, for

1 =< t =< s + 1 we can write the actual demand for period s as

Ds =

Ds at the beginning of period t1.

Let Et[Ds] and Vart[Ds] denote the expectation and the variance of Ds given what is

known at the beginning of period t.

Thus,

is just the unbiased forecast

is a measure of the forecast error

Questions?

If we apply (MMFE) to Autoregressive Integrated Moving Average (ARIMA) Demand Model.

D1 = µ + for s >=2

Ds = µ +

where 's are zero mean random variables with variance . is constant and is between 0 and 1.

- For (ARIMA), knowing that for all s > t, and , how can I prove that this model (ARIMA) can be fit into the framework of (MMFE)?

- In the formula Ds = µ + does each , , ....and are independent of each other and all have the same variance of ?

I need to find the variance.

What is the variance (if it is too much work to find the variance, it is okay to leave it)?

#### Solution Preview

See the attached sheet. I hope you understand the arguments.

Using the martingale method of forecast evolution (MMFE) of heath and Jackson and obtain a formula for the mean and variance of the lead time demand and apply these formulas to specific demand models.

The Martingale Method of Forecast Evolutions (MMFE) can be represented as follows:

At the beginning of the first period, there is an initial forecast for the demand to prevail

in period s for each s >= 1 in the planning horizon. Forecasts are updated at the beginning

of every period as follows: For all s >= t

+

where is the forecast at the beginning of period t for the demand to prevail ...

#### Solution Summary

The martingale method of forecast evolutions of health are determined. The mean and variance of the lead time demands are given.