Math 6010 • Linear Models

Suppose \(N\) is a positive random variable.
\begin{parts}
\item Show that
\[
P(N>0)\geq \frac{(EN)^2}{E N^2}.
\]
(5 points)
\item Let \(X_1,X_2,\ldots\) be i.i.d. random variables. Define \(S_n=X_1+\cdots+X_n\). Use part (a) to show that for any positive integer \(n\),
\[
P(S_j=0\text{, for some }1\leq j\leq n)
\geq
\frac{\sum_{j=1}^{n}P(S_j=0)}
{2\left(1+\sum_{j=1}^{n}P(S_j=0)\right)}.
\]
(5 points)
\end{parts}

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{\large\bfseries Statistics}
\end{center}