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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 1

Question

Math 6010 • Linear Models Fall 2009 • Question 1

Let \(X\) be an exponential random variable with mean 1, that is, \(\mathbb {P}(X>u)=e^{-u}\) for all \(u>0\). Evaluate \(\mathbb {E}[X\mid X\wedge t]\) and \(\mathbb {E}[X\mid X\vee t]\) for all \(t>0\). Here \(a\wedge b:=\min (a,b)\) and \(a\vee b:=\max (a,b)\).

Raw LaTeX

Let \(X\) be an exponential random variable with mean 1, that is,
\(\Prob(X>u)=e^{-u}\) for all \(u>0\). Evaluate \(\E[X\mid X\wedge t]\) and
\(\E[X\mid X\vee t]\) for all \(t>0\). Here
\(a\wedge b:=\min(a,b)\) and \(a\vee b:=\max(a,b)\).

Topic Tags

Exponential Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods