Ralph has $\$500$ in the bank when he decides to try a savings experiment. On each day $i \in [1;30]$, Ralph flips a fair coin. If it comes up heads, he deposits $i$ dollars into the bank; if it comes up tails, he withdraws $\$10$. How many dollars should he expect to have in the bank after $30$ days?
I'm a bit unsure of how to approach this problem. Should I calculate the expected value of dollars saved each day till day $30$ and then sum them all up. For instance, the expected value of day $1$ is $(1/2 * 1) - (1/2 * 10)$, the expected value of day $2$ is $(1/2 * 2) - (1/2 * 10)$, etc. My thinking is that by calculating the expected savings of each day, I can add the sum of those values to $\$500$ to obtain the expected money Ralph expects to have in the bank? Any help would be great!
As you suggest, you can just sum the expected changes on each day. The expected change in savings on day $i$ are $\frac 12(i-10)$. Thus at the end of the month you expect to have $$500+\frac 12\times \sum_{i=1}^{30} (i-10)=\frac {1165}2=582.5$$