Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 6

Question

Math 6010 • Linear Models Fall 2015 • Question 6

Let \(X_1,X_2\) be independent and identically distributed exponential\((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Find the joint density function of \[ Y_1=X_1+X_2 \quad \text {and}\quad Y_2=\frac {X_1}{X_1+X_2}. \]

Raw LaTeX

Let \(X_1,X_2\) be independent and identically distributed
exponential\((\lambda)\) random variables, i.e. the common density is
\[
f(t)=
\begin{cases}
0, & \text{if } t<0,\\
\dfrac{1}{\lambda}\exp(-t/\lambda), & \text{if } t\geq0.
\end{cases}
\]
Find the joint density function of
\[
Y_1=X_1+X_2
\quad\text{and}\quad
Y_2=\frac{X_1}{X_1+X_2}.
\]

Topic Tags

Covariance and Independence Exponential Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods