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Author Archives: mzspivey
Pascal Matrices and Binomial Inversion
In this post we’ll look at the relationship between a Pascal matrix, its inverse, and binomial inversion. It turns out that these are the same concepts viewed from two different angles. The Pascal matrix is the matrix containing Pascal’s triangle … Continue reading
Posted in binomial coefficients, matrices
2 Comments
Counting Poker Hands
For this post I’m going to go through a classic exercise in combinatorics and probability; namely, proving that the standard ranking of poker hands is correct. First, here are the standard poker hands, in ranked order. Straight flush: Five cards of … Continue reading
Posted in combinatorics, probability
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An Expected Value Connection Between Order Statistics from a Discrete and a Continuous Distribution
Years ago, in the course of doing some research on another topic, I ran across the following result relating the expected values of the order statistics from a discrete and a continuous distribution. I found it rather surprising. Theorem: Fix n, and … Continue reading
Posted in order statistics, probability, statistics
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Two Methods for Proving an Alternating Binomial Identity
Recently I gave an optional homework problem in which I asked students to prove the following binomial identity: . (Here, I’m using the Iverson bracket notation in which if P is true and if P is false.) I intended for students to … Continue reading
Posted in binomial coefficients, combinatorics
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A Bonus Question on Convergent Series
Occasionally when teaching the sequences and series material in secondsemester calculus I’ve included the following question as a bonus: Question: Suppose is absolutely convergent. Does that imply anything about the convergence of ? The answer is that converges. I’m going … Continue reading
Posted in calculus, sequences and series
6 Comments
My Experiences on a PostElection Panel
No mathematics post this month. Instead, I’m just going to link to an article I published last week in Inside Higher Ed. In this article I describe my experiences as the conservative voice on a panel held on my campus on November 9. This was … Continue reading
Posted in campus issues
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Relations That Are Symmetric and Antisymmetric
When I teach relations and their properties, the question of whether a relation can be both symmetric and antisymmetric always seems to arise. This post addresses that question. First, a reminder of the definitions here: A relation on a set … Continue reading
Posted in relations
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