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Binomial Theorem

Can you expand on $(x+y)^{2}$? I guess you would find that is quite easy to do. You can easily find that $(x+y)^{2} = x^{2}+ 2xy +y^{2}$.

How about the expansion of $(x+y)^{10}$. It is no longer easy.

It is no longer easy, isn’t it. However, if we use Binomial Theorem, this expansion becomes an easy problem.

Binomial Theorem is a very intriguing topic in mathematics and it has a wide range of applications.

Theorem

Let $x$, $y$ be real numbers (or complex, or polynomial). For any positive integer $n$, we have:

theorem

where,

theorem

Proof:

We will use prove by induction. The base case $n=1$ is obvious. Now suppose that the theorem is true for the case $n-1$, that is assume that:

theorem

 

we will need to  show that, this is true for

theorem

Let us consider the left-hand side of the equation above

theorem

We can now apply Pascal’s identity:

 

Pascal's identity

The equation above can be simplified to:

Pascal's identity

as we desired.

Example 1:  Power rule in Calculus

 

In calculus, we always use the power rule that Power rule

 

We can prove this rule using the Binomial Theorem.

Proof:

Recall that derivative for any continuous function f(x) is defined as:

 

Binomial Theorem

Let $n$ be a positive integer and let $f(x) = x^{n}$

 

The derivative of f(x) is:

 

Binomial Theorem

Example 2:  Binomial Distribution 

Let X be the number of Head a sequence of n independent coin tossing. X is usually model by binomial distribution in the probability model. Let $ p \in [0,1]$ be the probability that a head shows up in a toss, and let $k = 0,1,\dots,n$. The probability that there is $k$ head in the sequence of $n$ toss is:

Binomial Distribution

We know that sum of all the probability must equal to 1. In order to show this, we can use Binomial Theorem. We have:

 

Binomial Distribution

Please also check another article Gaussian Samples and N-gram language models ,Bayesian model, Monte Carlo for statistics knowledge.

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