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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 7

Question

Math 6010 • Linear Models Fall 2013 • Question 7

Suppose that, for every integer \(n\geq 1\), \(X(n)\) has a Gamma distribution with parameters \(n+1\) and 1; that is, the probability density function of \(X(n)\) is \[ f_n(x):=\frac {x^ne^{-x}}{n!}\mathbf {1}_{(0,\infty )}(x). \]

(a) Compute \(\mathbb {E}(X(n))\) and \(\operatorname {Var}(X(n))\) for every integer \(n\geq 1\).

(b) Prove that, as \(n\to \infty \), \[ \frac {X(n)-n}{\sqrt {n}}\Rightarrow \mathrm {N}(0,1). \]

Raw LaTeX

Suppose that, for every integer \(n\geq1\), \(X(n)\) has a Gamma
distribution with parameters \(n+1\) and 1; that is, the probability density
function of \(X(n)\) is
\[
  f_n(x):=\frac{x^ne^{-x}}{n!}\mathbf{1}_{(0,\infty)}(x).
\]

\textbf{(a)} Compute \(\E(X(n))\) and \(\Var(X(n))\) for every integer \(n\geq1\).

\newpage

\textbf{(b)} Prove that, as \(n\to\infty\),
\[
  \frac{X(n)-n}{\sqrt{n}}\Rightarrow \mathrm{N}(0,1).
\]

Topic Tags

Central Limit Theorems Measure-Theoretic Probability Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods