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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 7

Question

Math 6010 • Linear Models Spring 2020 • Question 7

Let \(X\) and \(Y\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\theta }e^{-t/\theta }, & \text {if } t\geq 0. \end {cases} \] Compute the density function of \[ Z=\frac {X}{X+Y}. \]

Raw LaTeX

Let \(X\) and \(Y\) be independent identically distributed random
variables with density function
\[
  f(t,\theta)=
  \begin{cases}
    0, & \text{if } t<0,\\
    \dfrac{1}{\theta}e^{-t/\theta}, & \text{if } t\geq 0.
  \end{cases}
\]
Compute the density function of
\[
  Z=\frac{X}{X+Y}.
\]

Topic Tags

Covariance and Independence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods