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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 8

Question

Math 6010 • Linear Models Fall 2015 • Question 8

Let \(X\) and \(Y\) be independent random variables with densities \(f\) and \(g\), where \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{2}\exp (-t/2), & \text {if } t\geq 0, \end {cases} \] and \[ g(t)= \begin {cases} 0, & \text {if } t\notin [-1,2],\\ \dfrac {1}{3}, & \text {if } t\in [-1,2]. \end {cases} \] Compute the density of \(X+Y\).

Raw LaTeX

Let \(X\) and \(Y\) be independent random variables with densities
\(f\) and \(g\), where
\[
f(t)=
\begin{cases}
0, & \text{if } t<0,\\
\dfrac{1}{2}\exp(-t/2), & \text{if } t\geq0,
\end{cases}
\]
and
\[
g(t)=
\begin{cases}
0, & \text{if } t\notin[-1,2],\\
\dfrac{1}{3}, & \text{if } t\in[-1,2].
\end{cases}
\]
Compute the density of \(X+Y\).

Topic Tags

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