Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 6

Question

Math 6010 • Linear Models 1993 • Question 6

Let \(U_1,U_2,\ldots ,U_n\) be independent identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\le U_{2,n}\le \cdots \le U_{n,n}\) denote the order statistics. Let \(X_1,X_2,\ldots ,X_{n+1}\) be independent identically distributed exponential random variables with \(\mathbb {E} X_1=1\). Define \(S(i)=X_1+\cdots +X_i\).

(a) Prove that the random vectors \[ \{U_{1,n},\ldots ,U_{n,n}\} \quad \text {and}\quad \left \{\frac {S(1)}{S(n+1)},\ldots ,\frac {S(n)}{S(n+1)}\right \} \] have the same distribution.

(b) Compute the asymptotic distribution of \(n(U_{i+3,n}-U_{i,n})\), as \(n\to \infty \) when \(i\) is fixed.

Let \(U_1,U_2,\ldots,U_n\) be independent identically distributed
random variables, uniform on \([0,1]\). Let
\(U_{1,n}\le U_{2,n}\le\cdots\le U_{n,n}\) denote the order statistics. Let
\(X_1,X_2,\ldots,X_{n+1}\) be independent identically distributed exponential
random variables with \(\E X_1=1\). Define \(S(i)=X_1+\cdots+X_i\).

(a) Prove that the random vectors
\[
\{U_{1,n},\ldots,U_{n,n}\}
\quad\text{and}\quad
\left\{\frac{S(1)}{S(n+1)},\ldots,\frac{S(n)}{S(n+1)}\right\}
\]
have the same distribution.

(b) Compute the asymptotic distribution of \(n(U_{i+3,n}-U_{i,n})\), as
\(n\to\infty\) when \(i\) is fixed.

Topic Tags

Covariance and Independence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods