We have two collections of random variables $X_i$ and $Y_i$. The $X_i$ are independent and identically distributed with expectation $1$, and the $Y_i$ are also independent and identically distributed with expectation $2$. We first flip a coin once, and we define, for all $n$, a random variable $S_n$ by $S_n=\sum_iX_i$ if heads comes up, and $S_n = \sum_iY_i$ if tail comes up. Show that
(a) $E(\frac{S_n}{n})=\frac{3}{2}$.
(b) $P\left(\left|\frac{S_n}{n} − \frac{3}{2}\right| > \frac{1}{4}\right)$ does not converge to $0$, when $n \rightarrow \infty$.
(c) Why does (b) not contradict the law of large numbers?
I was able to show a, but I'm not sure on how to show $b$ and $c$. Any help would be wonderful!
You can consider the special case $X_i \equiv 1,Y_i \equiv 2$. Then $S_n/n$ is either $1$ or $2$ with probability $1$. But there is no contradiction because $S_n$ is not itself a sum of iid random variables.