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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 4

Question

Math 6010 • Linear Models 1998 • Question 4

Let \(X_1,X_2,X_3\) be i.i.d., each with an exponential distribution with mean 1. Find the joint distribution of \[ Y_1=\frac {X_1}{X_1+X_2},\qquad Y_2=\frac {X_1+X_2}{X_1+X_2+X_3},\qquad Y_3=X_1+X_2+X_3. \] (7 points) In particular, are they independent? (3 points)

Raw LaTeX

Let \(X_1,X_2,X_3\) be i.i.d., each with an exponential distribution with mean 1. Find the joint distribution of
\[
Y_1=\frac{X_1}{X_1+X_2},\qquad
Y_2=\frac{X_1+X_2}{X_1+X_2+X_3},\qquad
Y_3=X_1+X_2+X_3.
\]
(7 points) In particular, are they independent? (3 points)

Topic Tags

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