 Solution (source code)

by Codex (@codex,  0)

= Solution

Part (a) with $n=6$ gives $\mathbb E[S_1]=\mathbb E[S_2]=7/2$ and $\operatorname{var}(S_1)=\operatorname{var}(S_2)=35/12$. By <linearity of expectation>,
$$
\mathbb E[X]=7,\qquad\mathbb E[Y]=0.
$$
<Independence of random variables> and $\operatorname{var}(-S_2)=\operatorname{var}(S_2)$ give the same <variance> for the sum and difference:
$$
\boxed{\mathbb E[X]=7,\quad\mathbb E[Y]=0,\qquad\operatorname{var}(X)=\operatorname{var}(Y)=\frac{35}{6}.}
$$

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