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Intuitive Interpretation of Unexpected Binomial Coefficients

This post will give an intuitive interpretation of the presence of the combination formula (which equals the binomial coefficient) in math problem solutions and probability distributions that are seemingly unrelated to combinations. For example, the binomial coefficient shows up in the probability mass function of the binomial distribution and the negative binomial distribution.

Wikipedia has the following definition for combination:

In mathematicscombination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, an apple, orange and pear say, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally a kcombination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

The binomial coefficient indexed by n and k is denoted \tbinom nk. The formula below, for evaluating binomial coefficients, uses factorials.

\binom nk = \frac{n!}{k!\,(n-k)!} \quad \mbox{for }\ 0\leq k\leq n

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Probability Distributions Reference Table

Table Description

I have uploaded a reference table that I created with common probability distributions. Although the same formulas and similar descriptions can be obtained on Wikipedia, I find it useful to have the information contained on one table. The table does not have charts that plot probability density or probability mass functions (PDFs and PMFs) or cumulative distribution functions (CDFs). These charts can be found on Wikipedia or plotted with graphing software using the equations on the table. I find it helpful to look at PDF and CDF plots when learning about the distributions. There might be some closed form CDFs, moment generating functions, or other equations that I omitted from the table. Also, some distributions have various formulations. For example, the four distributions defined below are all very similar, and I imagine that each might be referred to as a negative binomial, although certain specifications are more conventional than others.

  1. A distribution of the number of failures in a sequence of Bernoulli trials, before a specified number of successes.
  2. A distribution of the number of successes in a sequence of Bernoulli trials, before a specified number of failures.
  3. A distribution of the number of successes and failures in a sequence of Bernoulli trials, until a specified number of successes.
  4. A distribution of the number of successes and failures in a sequence of Bernoulli trials, until a specified number of failures.

In cases where there are multiple ways to define a distribution, I either used the definition that I prefer, or gave multiple formulations of the distribution. I am including a link to both a PDF of the table and an Excel file that has the table and a VBA macro that centers images across a column or multiple columns. Also included is a link to the MathML markup file that was used to generate the equations on the table. Excel’s equation editor was giving me problems, and I did not want to type \LaTeX code manually, so I searched for a what you see is what you get (WYSIWYG) equation editor on the internet. I ended up finding an open source project, Formulator MathML Editor (http://code.google.com/p/formulator-mathml/), which I think is great. After getting started, I also found a \LaTeX editor, LyX (http://www.lyx.org/), which probably would have worked just as well, with the advantage that I’m more familiar with \LaTeX than MathML. I have spotted and corrected a few errors that I made when creating the table (e.g., I originally had the continuous uniform distribution’s variance equal to \frac{1}{12}\left(b+a\right)^{2}, which was corrected to \frac{1}{12}\left(b-a\right)^{2}), but there still may be some errors. Please let me know if you find any.

Links

PDF: Probability-Distributions.pdf
Excel: Probability-Distributions.xlsm
MathML: Distribution-Formulas.mml