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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 5

Question

Math 6010 • Linear Models Fall 2008 • Question 5

Let \(X\) and \(Y\) be independent exponential random variables with parameters \(\lambda \) and \(\mu \). (For example, \(\mathbb {P}(X>x)=e^{-\lambda x}\) for \(x>0\).)

(a) Show that \(Z:=\min (X,Y)\) is independent of the event \(\{X<Y\}\).

(b) Find \(\mathbb {P}(X=Z)\).

(c) Find the distribution of \(U:=(X-Y)^+=\max (X-Y,0)\).

(d) Find the distribution of \(V:=\max (X,Y)-\min (X,Y)\).

Raw LaTeX

Let \(X\) and \(Y\) be independent exponential random variables with
parameters \(\lambda\) and \(\mu\). (For example, \(\Prob(X>x)=e^{-\lambda x}\) for
\(x>0\).)

(a) Show that \(Z:=\min(X,Y)\) is independent of the event \(\{X<Y\}\).

(b) Find \(\Prob(X=Z)\).

(c) Find the distribution of \(U:=(X-Y)^+=\max(X-Y,0)\).

(d) Find the distribution of \(V:=\max(X,Y)-\min(X,Y)\).

Topic Tags

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