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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 1

Question

Math 6010 • Linear Models Spring 2009 • Question 1

Let \(X_1,X_2,\ldots \) be i.i.d. with mean \(\mu \) and variance \(\sigma ^2\), and put \(S_0:=0\) and \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Let \(N\) be a nonnegative-integer-valued random variable independent of \(X_1,X_2,\ldots \). Find the mean and variance of \(S_N\) by conditioning on \(N\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. with mean \(\mu\) and variance
\(\sigma^2\), and put \(S_0:=0\) and \(S_n:=X_1+\cdots+X_n\) for each \(n\geq1\). Let
\(N\) be a nonnegative-integer-valued random variable independent of
\(X_1,X_2,\ldots\). Find the mean and variance of \(S_N\) by conditioning on \(N\).

Topic Tags

Analysis of Variance Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods