Exercise :
We toss two dices. Let $X$ be the indication of the result of the first one and $Y$ the indication of the result of the second one. If $S=X+Y$, find the following :
(a) The probability mass function $p(s_j)$ of the random variable $S$.
(b) The mean value $E[S]$ of the random variable $S$.
(c) The variance $V[S]$ of the random variable $S$.
Attempt :
(a)
So, $S$ is the summation of the indications of the two dices, which means that the value of $S$ will rely in $[2,12]$, which is obvious. How would one continue to finding the probability mass function though ?
For (b) and (c), I do not know how to continue (I think you need (a) for those as well).
This is not a homework question, as it is an exam question which I'm trying to figure out in order for the upcoming semester tests. I would really appreciate any help on understanding the problem and the questions asked.
Hint:
The pdf of $X$ (or Y) is a uniform distribution on support set $\{1,2,3,4,5,6\}$ so the distribution of $S$ using convolution operator over its support set $\{2,3,4,5,6,7,8,9,10,11,12\}$ is $$\dfrac{1}{36},\dfrac{2}{36},\dfrac{3}{36},\dfrac{4}{36},\dfrac{5}{36},\dfrac{6}{36},\dfrac{5}{36},\dfrac{4}{36},\dfrac{3}{36},\dfrac{2}{36},\dfrac{1}{36}$$Can you finish now?