Conditional expectation of sum of dice rolls

We roll a dice once, if the result $X_1=1$ we stop, in other case we roll dice $X_1 -1$ more times, so the total number of roll is $X_1$. $S$ is the sum of obtained results, including the first one. What is $\mathbb{E}(S|X_1)$?

• did you try to write it as a sum of conditional probabilities, conditioned on the first roll and use that to calculate the full probability law. The possible values are $\{6,7,8,\dots,36\}$. – Gregory Grant Feb 23 '15 at 15:13

Preassuming that it is a six-sided fair die: $$\mathbb E(S\mid X_1=k)=k+(k-1)\times3\frac12$$
• Actually I computed the mean $\frac 1 6 \sum_1^6(X_i +(X_i -1)\mathbb{E}(X_i))P(X_i)= (\mathbb{E}(X_i) + \mathbb{E}(X_i)^2 -\mathbb{E}(X_i))= \mathbb{E}(X_i)^2= 12,25$, which is similar to R simulation I done to check it. – Daniel Borek Feb 23 '15 at 15:44
• You are right, I didn't use here the information about $X_1$ – Daniel Borek Feb 23 '15 at 15:55