was revising my stuffs for my stochastics exams and came across this question that I couldn't figure my way around..
Let $X_1,\ldots,X_n$ be independent, identically distributed random variables with $E(X_1) =a$ and
$S_n = \frac{1}{n} \sum_{i = 1}^n X_i$
Using the Chebychev inequality, give the smallest possible value of $x$, where $\mathbb{P}(\left | S_{100} - a \right | \geq x) \leq 0.01$, in the case where $X_{1}$ with the parameters $p \in [0,1]$ is:
a) binomially distributed ($B(10,p)$)
b) geometrically distributed
I gathered that $0.01 =\frac{Var(X))}{x^2}$ owing to the Chebychev inequality and figured that if I could derive $Var(X)$, I could then get $x$. And using the given distributions (binomial or geometric), I attempted to solve for $Var(X) = E(X^2) - (E(X))^2$, although I couldn't proceed any further with the finding of $E(X^2)$.
Although first and foremost, am I heading off into the right track? If so, how should I go about finding $E(X^2)$? Thanks for the help, as always!