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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 6

Question

Math 6010 • Linear Models 2000 • Question 6

\(X\) and \(Y\) are two independent identically distributed normal \(N(0,\sigma ^2)\) random variables. Let \[ Z=X^2+Y^2 \] and \[ W=\frac {X}{(X^2+Y^2)^{1/2}}. \] (a) Show that \(Z\) and \(W\) are independent.

(b) Compute the marginal distributions of \(Z\) and \(W\).

Raw LaTeX

\(X\) and \(Y\) are two independent identically distributed normal \(N(0,\sigma^2)\) random variables. Let
\[
Z=X^2+Y^2
\]
and
\[
W=\frac{X}{(X^2+Y^2)^{1/2}}.
\]
(a) Show that \(Z\) and \(W\) are independent.

(b) Compute the marginal distributions of \(Z\) and \(W\).

Topic Tags

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