I am curretly going through some expected value problems and I got stuck on this question:
(a) i) Consider the random variable X, based on rolling a fair die: If the outcome is 1,2, 3, 4, or 5, then this is the result. If it is 6, then the die is rolled again and the outcome added to the 6 already obtained. What is the expected value of X?
ii) Now consider the variable Y, which is realised like X, except that whenever the outcome is 6 the die is rolled again, and the result is the sum of all sixes plus the final outcome (which must be different from 6). What is the expected value of Y?
I calculated the expected value for part i) but I do not now how to calculate the second part as I am not sure how to deal with the fact that it can go to infinity.
Any advice is appreciated.
For the first, you have $\frac 16$ chance of rolling a $6$ on the first roll, then you get a second roll to make the expectation $9.5$. Otherwise you have $\frac 56$ chance not to roll a $6$ with expectation $3$. $$E(X)=\frac 16\cdot 9.5+\frac 56\cdot 3=\frac {49}{12}$$
For the second part, note that the expected addition after rolling a $6$ is the same as the expectation at the start, while the expectation if you don't roll a $6$ is $3$, so $$E(X)=\frac 16(6+E(X))+\frac 56 \cdot 3\\\frac 56E(X)=\frac 72\\E(X)=\frac {21}5$$