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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 1

Question

Math 6010 • Linear Models Spring 2010 • Question 1

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with distribution function \[ F(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ t^3, & \text {if } 0\leq t\leq 1,\\ 1, & \text {if } t>0. \end {cases} \] Show that \[ Y_n=n^{1/3}X_{1,n} \] converges in distribution, where \(X_{1,n}=\min \{X_1,X_2,\ldots ,X_n\}\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed
random variables with distribution function
\[
F(t)=
\begin{cases}
0, & \text{if } -\infty<t<0,\\
t^3, & \text{if } 0\leq t\leq 1,\\
1, & \text{if } t>0.
\end{cases}
\]
Show that
\[
Y_n=n^{1/3}X_{1,n}
\]
converges in distribution, where
\(X_{1,n}=\min\{X_1,X_2,\ldots,X_n\}\).

Topic Tags

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