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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 2

Question

Math 6010 • Linear Models Spring 2016 • Question 2

(a) Find a simple asymptotic formula for \(\mathbb {P}(X>x)\), where \(X\) has the GAMMA\((\theta ,\alpha )\) density, \[ f(x)=\frac {1}{\theta ^{\alpha }\Gamma (\alpha )} x^{\alpha -1}e^{-x/\theta }, \qquad x>0. \]

(b) Let \(X_1,X_2,\ldots \) be i.i.d. GAMMA\((\theta ,\alpha )\). Find an increasing sequence \(a_n\) such that \(\limsup _{n\to \infty }X_n/a_n=1\) a.s., and prove it.

Raw LaTeX

\textbf{(a)} Find a simple asymptotic formula for \(\Prob(X>x)\),
where \(X\) has the GAMMA\((\theta,\alpha)\) density,
\[
  f(x)=\frac{1}{\theta^{\alpha}\Gamma(\alpha)}
  x^{\alpha-1}e^{-x/\theta}, \qquad x>0.
\]

\textbf{(b)} Let \(X_1,X_2,\ldots\) be i.i.d. GAMMA\((\theta,\alpha)\). Find an
increasing sequence \(a_n\) such that \(\limsup_{n\to\infty}X_n/a_n=1\) a.s., and
prove it.

Topic Tags

Convergence in Probability and Almost Sure Convergence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods