Q:

Expected Value of Dollars in Bank

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!

A:

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 $$500+\frac 12\times \sum_{i=1}^{30} (i-10)=\frac {1165}2=582.5$$

