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Author Archives: mzspivey
The Divergence of the Harmonic Series
The fact that the harmonic series, , diverges has been known since the time of Nicole Oresme in the 14th century, but this fact is still somewhat surprising from a numerical standpoint. After all, each successive term only adds a … Continue reading
Posted in harmonic numbers, sequences and series
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The Secretary Problem With the Two Best
The secretary problem is the following: Suppose a manager wants to hire the best person for his secretary out of a group of n candidates. He interviews the candidates one by one. After interviewing a particular candidate he must either (1) … Continue reading
Posted in optimization, probability
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Some Thoughts on Teaching Math for Social Justice
Short version: I’m not in favor of it. For a teacher to use their position of power to push their political and social views on their students is an abuse of that power. Long version: See the rest of this … Continue reading
Posted in campus issues, politics, social justice, teaching
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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
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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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