# 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 = Âµ + for s >=2

Ds = Âµ +

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 = Âµ + 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)?

https://brainmass.com/business/business-management/random-variable-functions-evolutions-11936

## SOLUTION This solution is **FREE** courtesy of BrainMass!

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 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 = Âµ + for s >=2

Ds = Âµ +

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)?

If we know that for all s>t , and , taking a careful look at MMFE and ARIMA Models , they have the same framework.

ARIMA - = Âµ + , WE KNOW THAT Âµ is an estimate of Ft,s ( Note Âµ

Is the mean expected demand over lead time which is precisely what Ft,s tells us., the unbiased forcast ). So the Arima model can be rewritten as :

Fts + ...now looking at the second part , we can write it as

(beta*E1, + beta*E2,.........beta*Es-1) and we have been told that so it can re written as

(Et,s -1 +Et+1,s-1......Es-1,s-1).....ALSO SO that would mean that we will have summationof Ej,s from j=t to s)...this is precisely the argument in the MMFS.

Note: I am not very proficient with word, hope you can make out my arguments. I am trying to arrange a scanner so that I can mail you the procedure when I do it step by step by hand.

- In the formula Ds = Âµ + 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)?

Yes in this each of them are independent of each other and have the same variance of .

Mail me if you don't understand the Explanation.

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