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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 1

Question

Math 6010 • Linear Models 1994 • Question 1

Let \(X_1,X_2,\cdots \) be independent, identically distributed random variables, uniformly distributed on \([0,1]\). Show that \[ Y_n=\frac {4\sum _{1\le k\le n}kX_k-n^2}{n^{3/2}}, \] converges in distribution to a normal random variable.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent, identically distributed random variables, uniformly distributed on \([0,1]\). Show that
\[
Y_n=\frac{4\sum_{1\leq k\leq n} kX_k-n^2}{n^{3/2}},
\]
converges in distribution to a normal random variable.

Topic Tags

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