You are a division manager who submits a budget to corporate headquarters $($HQ$)$ at the start of each month. Based on that budget, HQ sends you the resources you request.

Revenues are expected to be $$600,000$ for the month.

Costs will be somewhere between $$400,000$ and $$600,000($inclusive$).$ Moreover, the costs are uniformly distributed, meaning that each potential realization of actual costs between $$400,000$ and $$600,000$ is equally likely.

Before you submit your budget, you know for certain what your division$$s monthly costs will be$.$ However, HQ only knows that costs will be somewhere between $$400,000$ and $$600,000($inclusive$),$ and that the costs are uniformly distributed in that range.

HQ earns profits equal to $$600,000$ less the resources you request in your budget.

You get to keep the difference between the amount of resources you request in your budget and the actual costs for the month.

Suppose you learn that actual costs for the month are $$500,000.$

What amount of resources do you request in your budget $($the amount you$$ll get from HQ$)?$

Budget: $$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$