I'm sure this is a duplicate of something but it's easier to just type it out right now. Since you have a constant value for median income, you just need the percent of a fire response area that is each income to arrive at an average median income.
Make sure you have an area attribute in your two files that is separate from the automatically tracked one (if there is such - won't be in a shapefile) that is calculated on the entire shape.
Intersect your fire response area and income polygons. The result should be a set of polygons with the income levels cut up and identified by response area. Create a new field in that table to hold the percentage of total area. And if there isn't one, you need a new area field for the new, cut up shapes. Field calc that new field to divide the total area of the fire response zone (should be an attribute for each row) by the area of that actual piece.
Now create another new field and field calc it to be area percentage times median income. Finally, use Summary Statistics with fire response zone id as a CASE field and your percentage median field as a statistics field with a sum type. Finally, open that table and then add a final field to divide that sum by the frequency field. That should be the median income for the zone. You'll have to join that table back to the fire response zones (or use Join Field) based on ID to get the attribute on the shape.