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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 3

Question

Math 6010 • Linear Models 2005 • Question 3

Suppose \(X_{\alpha ,\beta }\) has a Gamma distribution with parameters \((\alpha ,\beta )\). That is, the density function of \(X_{\alpha ,\beta }\) is \[ f(x)=\frac {\beta ^\alpha }{\Gamma (\alpha )}x^{\alpha -1}e^{-\beta x}, \qquad x>0, \] where \(\Gamma (\alpha )=\int _0^\infty x^{\alpha -1}e^{-x}\,dx\) for all \(\alpha >0\). Prove that \((\beta X_{\alpha ,\beta }-\alpha )/\sqrt {\alpha }\Rightarrow N(0,1)\).

Raw LaTeX

Suppose \(X_{\alpha,\beta}\) has a Gamma distribution with parameters
\((\alpha,\beta)\). That is, the density function of \(X_{\alpha,\beta}\) is
\[
  f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},
  \qquad x>0,
\]
where \(\Gamma(\alpha)=\int_0^\infty x^{\alpha-1}e^{-x}\,dx\) for all \(\alpha>0\).
Prove that \((\beta X_{\alpha,\beta}-\alpha)/\sqrt{\alpha}\Rightarrow N(0,1)\).

Topic Tags

Estimator Distributions Multivariate Normal Distributions Quadratic Forms and Chi-Squared Distributions Uniform Integrability and Truncation Methods