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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 3

Question

Math 6010 • Linear Models 1998 • Question 3

Let \(X_1,X_2,\ldots \) be independent random variables with distribution functions \(F_1,F_2,F_3,\ldots \). Let \(Y=\sup _{1\le i<\infty }X_i\).

(a) Show that \(\mathbb {P}\{Y<\infty \}\) is 0 or 1 depending on whether \(\sum _{1\le i<\infty }(1-F_i(x))\) converges for some \(x\). (5 points)

(b) Show that if \(\mathbb {P}\{Y<\infty \}=1\), then \(\prod _{1\le i<\infty }F_i(x)\) converges for all \(x\) and it is the distribution function of \(Y\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent random variables with
distribution functions \(F_1,F_2,F_3,\ldots\). Let
\(Y=\sup_{1\le i<\infty}X_i\).

(a) Show that \(\Prob\{Y<\infty\}\) is 0 or 1 depending on whether
\(\sum_{1\le i<\infty}(1-F_i(x))\) converges for some \(x\). (5 points)

(b) Show that if \(\Prob\{Y<\infty\}=1\), then
\(\prod_{1\le i<\infty}F_i(x)\) converges for all \(x\) and it is the distribution
function of \(Y\).

Topic Tags

Covariance and Independence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods