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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 8

Question

Math 6010 • Linear Models Fall 2018 • Question 8

Prove the identity \[ \frac {\sin \theta }{\theta } = \int _0^1 e^{i\theta (2x-1)}\,dx. \] Then use this to show that \[ \frac {\sin \theta }{\theta } = \prod _{n=1}^{\infty }\cos (\theta /2^n). \] Hint: Use that a Uniform\((0,1)\) random variable can be written as an infinite weighted sum of iid Bernoulli\((1/2)\) random variables.

Raw LaTeX

Prove the identity
\[
  \frac{\sin\theta}{\theta}
  =
  \int_0^1 e^{i\theta(2x-1)}\,dx.
\]
Then use this to show that
\[
  \frac{\sin\theta}{\theta}
  =
  \prod_{n=1}^{\infty}\cos(\theta/2^n).
\]
\textbf{Hint:} Use that a Uniform\((0,1)\) random variable can be written as an
infinite weighted sum of iid Bernoulli\((1/2)\) random variables.

Topic Tags

Infinite Products and Multiplicative Martingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weighted Sums and Triangular Arrays