Expected Sum of n numbers after m random opeartions on 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 the objects. I have to do this $5$ times. What will be the expected sum of all objects after all $5$ turns?

Answer 1

I am letting $S_{i}$ indicate the random number from S in draw i. So we want to find $$E\left(\sum_{j=1}^{25}\left(j+\sum_{i=1}^{5}S_{i}\right)\right)=E\left(\sum_{j=1}^{25}j+25\sum_{i=1}^{5}S_{i}\right)=\sum_{j=1}^{25}j+25\sum_{i=1}^{5}E(S_{i})$$

where we note that $\sum_{j=1}^{25}j=\frac{25(26)}{2}=325$ so we have are expected value of sum will be

$$=325+25\sum_{i=1}^{5}E(S_{i})$$ where $$E(S_{i})=\frac{1}{5}(-2-1+0+1+2)=0$$ And since sampling with replacement this is true for i=1,..,5 so finally we have $$=325+25\sum_{i=1}^{5}0=325+25(0)=325$$

If anyone sees any mistakes or ways this can be done better please comment.