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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 1

Question

Math 6010 • Linear Models 2000 • Question 1

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential (1) random variables with common density function \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ e^{-t}, & \text {if } 0\leq t<\infty . \end {cases} \] (a) Show that \(\left (\sum _{1\leq i\leq n}X_i\right )^{1/2}-n^{1/2}\) converges in distribution to a normal random variable.

(b) Compute the mean and the variance of the limiting normal random variable.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed exponential (1) random variables with common density function
\[
f(t)=
\begin{cases}
0, & \text{if } -\infty<t<0,\\
e^{-t}, & \text{if } 0\leq t<\infty.
\end{cases}
\]
(a) Show that \(\left(\sum_{1\leq i\leq n}X_i\right)^{1/2}-n^{1/2}\) converges in distribution to a normal random variable.

(b) Compute the mean and the variance of the limiting normal random variable.

Topic Tags

Covariance and Independence Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits