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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 3

Question

Math 6010 • Linear Models Spring 2020 • Question 3

Let \(X\) be a non-negative, integer-valued random variable.

a) Prove that \(\mathbb {E}[X]=\sum _{n=1}^{\infty }\mathbb {P}(X\geq n)\).

b) A dresser has \(k\) distinct pairs of socks (so \(2k\) socks total) and the socks are unmatched. We select, at random and without replacement, one sock at a time until a pair has been drawn. Compute the expectation of the total number of draws needed.

Raw LaTeX

Let \(X\) be a non-negative, integer-valued random variable.

a) Prove that \(\E[X]=\sum_{n=1}^{\infty}\Prob(X\geq n)\).

b) A dresser has \(k\) distinct pairs of socks (so \(2k\) socks total) and the
socks are unmatched. We select, at random and without replacement, one sock at
a time until a pair has been drawn. Compute the expectation of the total number
of draws needed.

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Topic Tags

Conditional Expectation and Conditional Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods