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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 7

Question

Math 6010 • Linear Models Spring 2010 • Question 7

If \(X\) is either (a) Poisson\((\lambda )\) or (b) gamma\((\lambda ,1)\) (density proportional to \(x^{\lambda -1}e^{-x}\), \(x>0\)), show that \((X-\mathbb {E}[X])/\sqrt {\operatorname {Var}(X)}\) converges in distribution to \(N(0,1)\) as \(\lambda \to \infty \) (\(\lambda \) need not be an integer).

Raw LaTeX

If \(X\) is either \textbf{(a)} Poisson\((\lambda)\) or \textbf{(b)}
gamma\((\lambda,1)\) (density proportional to \(x^{\lambda-1}e^{-x}\), \(x>0\)),
show that \((X-\E[X])/\sqrt{\Var(X)}\) converges in distribution to \(N(0,1)\) as
\(\lambda\to\infty\) (\(\lambda\) need not be an integer).

Topic Tags

Estimator Distributions Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits