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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 \[ \text {(*)}\qquad \mathbb {E}\left [\left (\frac {S_n-n}{\sqrt n}\right )^-\right ]\to \mathbb {E}[Z^-]. \]

(b) Evaluate both expectations in (*) explicitly, and, noting the telescoping sum, deduce Stirling’s formula, for \(n!\).

Raw LaTeX

(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
\[
\text{(*)}\qquad
\E\left[\left(\frac{S_n-n}{\sqrt n}\right)^-\right]\to \E[Z^-].
\]

(b) Evaluate both expectations in (*) explicitly, and, noting the telescoping
sum, deduce Stirling's formula, for \(n!\).

Topic Tags

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