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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 5

Question

Math 6010 • Linear Models 1998 • Question 5

Suppose \(N\) is a positive random variable.

(a) Show that \[ \mathbb {P}(N>0)\ge \frac {(\mathbb {E} N)^2}{\mathbb {E} N^2}. \] (5 points)

(b) 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 \(n\), \[ \mathbb {P}(S_j=0,\text { for some }1\le j\le n) \ge \frac {\sum _{j=1}^n\mathbb {P}(S_j=0)} {2\left (1+\sum _{j=1}^n\mathbb {P}(S_j=0)\right )}. \] (5 points)

Statistics

Raw LaTeX

Suppose \(N\) is a positive random variable.

(a) Show that
\[
\Prob(N>0)\ge \frac{(\E N)^2}{\E N^2}.
\]
(5 points)

(b) 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 \(n\),
\[
\Prob(S_j=0,\text{ for some }1\le j\le n)
\ge
\frac{\sum_{j=1}^n\Prob(S_j=0)}
{2\left(1+\sum_{j=1}^n\Prob(S_j=0)\right)}.
\]
(5 points)

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\textbf{Statistics}
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Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods