Expected Sum of n numbers after m random opeartions on range of n numbers.

Question

I have been given 25 objects numbered from 1 to 25 and a set S = {-2, -1, 0, 1, 2}. I have to choose a random number from the given set and add it to all objects from 1 to 5. Again choose a random number and add it to objects from 15 to 25 and again choose a random number and add it to objects from 5 to 12, indexing of objects is 1 based.What will be the expected sum of all objects after all 3 turns?

Answer 1

Hints:

• What is the sum of what you started with?

• What is the expected amount you add?

• Hence, what is the expected sum of what you finish with?

See: linearity of expectation