A large retailer sells high fashion ski jackets and must commit to specific production quantities for all of its ski jackets several months in advance. Overestimating customer demand will result in unsold inventory while underestimating customer demand will lead to stockouts. The retailer has used historical data, and current economic conditions to construct the following probabilistic forecast of the demand for a particular style of ski jacket $($note that the forecasted demand is normally distributed$)$:

Demand$/$year $($units$)$ Probability

$5,00010\%$

$6,00010\%$

$7,00040\%$

$8,00020\%$

$9,00020\%$

The fixed production cost is $$100,000,$ the variable production cost per unit is $$150,$ the selling price during the ski season is $$350,$ and the salvage value for the ski jackets not sold during the ski season is $$100.$

Calculate the expected $($or average$)$ demand, based on the above probabilistic forecast. Please show all your work.