In a lottery, 10 tickets are drawn at random out of 50 tickets numbered from 1 to 50. What is the expected value of the sum of numbers on the drawn tickets?
My approach -
If after drawing a ticket, if it was replaced, then it can be easily solved.
Then, E(X) = 1*(1/50) + 2*(1/50) + ..... + 50*(1/50) = 51/2
However, I'm assuming that the tickets are not replaced because replacing a ticket does not make any sense. So, in that scenario, the expected value of any event (except the first one) is dependent on the expected values of the previous events. I can't figure out how to find expectation in this case or would it be the same as (51/2)*50 and I'm missing something.
Here is a trick that is useful when finding the expectation of a sum: expectation is linear.
Specifically, let $X_i$ be the value of the $i$th ticket you draw. Then $$ E(X_1+X_2 + \cdots +X_{10}) = E(X_1) + E(X_2) + \cdots + E(X_{10}) $$ In other words, while the probability distribution of the random variable $Y = X_1+X_2 + \cdots +X_{10}$ cares that you don't replace tickets, and this can make calculations complicated, expectation doesn't care. So it can still be as easily solved.