Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 6

Question

Math 6010 • Linear Models 2005 • Question 6

Suppose \(Y,X_1,X_2,\ldots \) have the following properties:

(a) \(Y\) has the exponential distribution. That is, \(\mathbb {P}\{Y>t\}=e^{-t}\) for \(t>0\).

(b) Conditionally on \(Y\), \(X_1,\ldots ,X_n\) are i.i.d. exponentials with parameter \(Y\), i.e., for all \(t_1,\ldots ,t_n>0\), \[ \mathbb {P}(X_1>t_1,\ldots ,X_n>t_n\mid Y) =\prod _{i=1}^{n}e^{-Yt_i} =e^{-Y\sum _{i=1}^{n}t_i} \qquad \text {a.s.} \]

Compute \(\mathbb {E}[Y\mid X_1,\ldots ,X_n]\). Use this to prove that \(\{M_n\}_{n=1}^{\infty }\) is a martingale, where \[ M_n=\frac {n+1}{1+X_1+\cdots +X_n} \qquad n=1,2,\ldots . \]

Suppose \(Y,X_1,X_2,\ldots\) have the following properties:

\textbf{(a)} \(Y\) has the exponential distribution. That is,
\(\Prob\{Y>t\}=e^{-t}\) for \(t>0\).

\textbf{(b)} Conditionally on \(Y\), \(X_1,\ldots,X_n\) are i.i.d. exponentials with
parameter \(Y\), i.e., for all \(t_1,\ldots,t_n>0\),
\[
  \Prob(X_1>t_1,\ldots,X_n>t_n\mid Y)
  =\prod_{i=1}^{n}e^{-Yt_i}
  =e^{-Y\sum_{i=1}^{n}t_i}
  \qquad \text{a.s.}
\]

Compute \(\E[Y\mid X_1,\ldots,X_n]\). Use this to prove that
\(\{M_n\}_{n=1}^{\infty}\) is a martingale, where
\[
  M_n=\frac{n+1}{1+X_1+\cdots+X_n}
  \qquad n=1,2,\ldots .
\]

Topic Tags

Conditional Expectation and Conditional Independence Convergence in Probability and Almost Sure Convergence Exponential Martingales Martingales and Supermartingales