How do you calculate the expected average of random values rounded to integers?

For example, consider following pseudocode:

round(random(0.3, 1.9))

A naive averaging of the two values comes out to 1.1, but the average of actual results is 0.84611.

How can that number be derived algorithmically, without actually generating a few thousand numbers?

Hopefully this isn't too trivial of a question, but I just can't seem to figure out a simple solution.