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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 10

Question

Math 6010 • Linear Models Spring 2016 • Question 10

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with characteristic function \(\varphi \). Let \(N\) be a random variable with \[ \mathbb {P}(N=k)=\frac {1}{2^k}, \qquad k=1,2,\ldots . \] We assume that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent.

(a) Compute the characteristic function of \(Y=X_1+X_2+\cdots +X_N\).

(b) Can you weaken the condition that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent and the formula obtained in (a) remains valid?

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent and identically distributed
random variables with characteristic function \(\varphi\). Let \(N\) be a random
variable with
\[
  \Prob(N=k)=\frac{1}{2^k}, \qquad k=1,2,\ldots .
\]
We assume that \(\{X_i,\ i\geq1\}\) and \(N\) are independent.

\textbf{(a)} Compute the characteristic function of
\(Y=X_1+X_2+\cdots+X_N\).

\textbf{(b)} Can you weaken the condition that \(\{X_i,\ i\geq1\}\) and \(N\) are
independent and the formula obtained in (a) remains valid?

Topic Tags

Characteristic Functions and Fourier Inversion Covariance and Independence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods