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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 11

Question

Math 6010 • Linear Models Fall 2009 • Question 11

Let \(X_1,X_2,\ldots \) be a sequence of i.i.d. random variables, each taking the values 0 and 1 with equal probabilities \(1/2\). Define \[ U:=\sum _{n=1}^{\infty }\frac {X_n}{2^n}. \] Prove that \(U\) is distributed uniformly on \((0,a)\) and compute \(a\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be a sequence of i.i.d. random variables, each
taking the values 0 and 1 with equal probabilities \(1/2\). Define
\[
  U:=\sum_{n=1}^{\infty}\frac{X_n}{2^n}.
\]
Prove that \(U\) is distributed uniformly on \((0,a)\) and compute \(a\).

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods