Proofs of Binomial and Geometric Series Expansions Under a Probabilistic Microscope with Real-Life Examples

This paper begins with a new and complete probabilistic proof of the well-known binomial theorem. Then, it broadens the same spirit for handling a useful infinite series of which the well-known geometric series forms a special case. Also included are applications of the theorems and real-life examples.


Introduction
While teaching probability and statistics, it has been my experience that both students and teachers feel more comfortable in employing probabilistic arguments to justify non-probabilistic mathematical conclusions. I have observed first-hand how probabilistic derivations of some well-known mathematical results tend to energize in my classrooms. More generally, such an approach may help in building a bridge between teaching and research at many levels by enriching both.
Professor Sujit K. Mitra, a legendary figure in statistics and mathematics, often quoted problems from linear algebra which he used to prove using probabilistic and statistical concepts. He had a unique style of teaching. One may find my interview piece (Mukhopadhyay 1997) with him interesting to read. I felt totally mesmerized every time Professor Mitra gave brilliant probabilistic arguments to make his students think outside the box.
I recall that Professor Mitra once quoted the following mathematical problem along with his ingenious probabilistic argument to prove it: Suppose Since a dispersion 150 ISSN-2424-6271 IASSL matrix must be p.s.d., the matrix B A  must be p.s.d. This proof is as crisp and pretty as one gets. It brings back memories and it continues to energize me to create probabilistic arguments to prove non-probabilistic mathematical conclusions when possible. This can be challenging and refreshing at the same time. Rosalsky (2007a) proved the binomial theorem using elementary probabilistic tools. Mukhopadhyay's (2007) letter to the editor and a response from Rosalsky (2007b) may be of interest to readers. Such a probabilistic treatment clearly remains important and it has drawn attention from other researchers recently. For example, a recent paper of Peterson (2013) testifies to this sentiment.
Binomial Theorem: For fixed real numbers b a, and a fixed positive integer n , the following holds: There is ample room to develop a complete proof of (1) via probabilistic arguments. In Section 2, I do so with a new and complete probabilistic proof. Then, in Section 3, I broaden the same spirit by handling a useful infinite series of which the well-known geometric series forms a special case. I include applications of the results briefly in Section 2.1 and Section 3.3. Additionally, real-life examples are given in Section 2.2 and Section 3.4. Suppose that n X X ,..., 1 are independent and identically distributed (i.i.d.)

Applications
Suppose that T is distributed as Binomial( p n, ). Then, we have: Also, we have:  .
These are well-known results that rely upon binomial theorem (1).

A Real-Life Example
Having observed a value t for T , one may estimate p by with its estimated bound of error: Under random sampling with replacement, the scenario will correspond to binomial sampling and I obtain from (4)- (5):  (5).
In survey sampling, however, it is more customary to gather observations under random sampling without replacement. If one assumes that one had observed the same responses under random sampling without replacement, then I should report: with its estimated bound of error: The extra expression (7) is referred to as the finite population correction. One may refer to Scheaffer et al. (2012, Eq (4.16), pp. 90-92). Hence, I immediately obtain from (6)- (7): Thus, with approximately 95% confidence, one may estimate p to lie between 50.7% and 69.3% which respectively correspond to B p  and B p  with B coming from (7).
The two sets of answers are very similar to each other because the sampling fraction, namely N n/ , is 0.1 which is rather small. That renders random sampling from a finite population, whether implemented using with replacement or without replacement, nearly equivalent. is referred to as the well-known geometric series. I will verify the infinite series expansion in (8) using only simple probabilistic steps in Section 3.1.

Probabilistic Derivation of an Infinite Series
The infinite series (8) One can see from (10) that in this infinite series expansion on the right-hand side, the 1 st term is 1, the 2 nd term is  r , the 3 rd term is , 1   (12).
The coefficient of 2   is:

An Alternative Derivation of (8) Under Case 2
Again, let me assume that 0

Remark 3.2.
In order to find the mean  and variance 2  of a negative binomial distribution or a geometric distribution, one customarily uses the result from (8) with appropriate positive integral values of r .

Applications
Let  Y denote a random variable that has the same distribution as that of Y defined by (9) corresponding to 1. = r Then, we have: in view of (8), which means Also, we have: These are well-known results that rely upon geometric series (8).

A Real-Life Example
When a large cyclone or hurricane lands and passes, the administration of forestry (filled largely with coconut trees) in a certain country may want to know the number of coconut trees which are still standing after the calamity. That is an important problem given the agricultural base of this country and its economic dependence on ample production of coconut.
However, it will be nearly impossible to count ( N = ) every single standing coconut tree. But, it will be possible to estimate the population size, N , by means The forest manager's team would go around, throughout the forest, ideally in all conceivable directions, and tie m yellow ribbons around the trunks of m standing coconut trees each of which is then called a marked or tagged coconut tree. When randomly sampled, if one observes a marked or tagged coconut tree, that will be recorded as a success, otherwise it will be recorded as a failure with Reusing the previous expressions from Section 3.3, after some calculations, I claim:

Epilogue
I have taught elementary probability and statistics courses for both undergraduate and first-year masters (MS) students from standard texts such as Scheaffer (1995), Ross (1997), Mukhopadhyay (2000), Casella and Berger (2001), and Wackerly et al. (2008) over many years. I have tested bits and pieces taken from what has been described in this unified paper in classrooms and during students' seminars as pilots with success. My students got a sense of what a research process might entail at their level. Such an approach helps in building a small bridge between teaching and research by enriching each other.