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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 7

Question

Math 6010 • Linear Models 2000 • Question 7

\(Z_1\) and \(Z_2\) are two independent identically distributed exponential \((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ \dfrac {1}{\lambda }e^{-t/\lambda }, & \text {if } 0\leq t<\infty . \end {cases} \] Let \(X=Z_1\) and \(Y=Z_1+Z_1Z_2\). Compute

(a) \(E(Y\mid X=x)\)

(b) \(\operatorname {var}(E(Y\mid X))\).

Raw LaTeX

\(Z_1\) and \(Z_2\) are two independent identically distributed exponential \((\lambda)\) random variables, i.e. the common density is
\[
f(t)=
\begin{cases}
0, & \text{if } -\infty<t<0,\\
\dfrac{1}{\lambda}e^{-t/\lambda}, & \text{if } 0\leq t<\infty.
\end{cases}
\]
Let \(X=Z_1\) and \(Y=Z_1+Z_1Z_2\). Compute

(a) \(E(Y\mid X=x)\)

(b) \(\operatorname{var}(E(Y\mid X))\).

Topic Tags

Covariance and Independence Exponential Martingales Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods