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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 2

Question

Math 6010 • Linear Models 1996 • Question 2
**(a)** Let \(X_1,X_2,\ldots\) be i.i.d. Poisson random variables with mean 1, and let \(Z\) be \(N(0,1)\). Denote \(a^-=-\min\{a,0\}\), and prove that \[ E\left[\left(\frac{S_n-n}{\sqrt{n}}\right)^-\right]\to E[Z^-]. \tag{*} \] **(b)** Evaluate both expectations in (*) explicitly, and, noting the telescoping sum, deduce Stirling's formula for \(n!\).

Raw LaTeX

\begin{parts}
\item Let \(X_1,X_2,\ldots\) be i.i.d. Poisson random variables with mean 1, and let \(Z\) be \(N(0,1)\). Denote \(a^-=-\min\{a,0\}\), and prove that
\[
E\left[\left(\frac{S_n-n}{\sqrt{n}}\right)^-\right]\to E[Z^-].
\tag{*}
\]
\item Evaluate both expectations in (*) explicitly, and, noting the telescoping sum, deduce Stirling's formula for \(n!\).
\end{parts}

Topic Tags

Central Limit Theorems Conditional Expectation and Conditional Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods