# Binomial Theorem

Can you expand on ? I guess you would find that is quite easy to do. You can easily find that .

How about the expansion of . 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  be real numbers (or complex, or polynomial). For any positive integer , we have: where, Proof:

We will use prove by induction. The base case is obvious. Now suppose that the theorem is true for the case , that is assume that: we will need to  show that, this is true for Let us consider the left-hand side of the equation above ## We can now apply Pascal’s identity: The equation above can be simplified to: as we desired.

### Example 1:  Power rule in Calculus

In calculus, we always use the power rule that We can prove this rule using the Binomial Theorem.

Proof:

Recall that derivative for any continuous function f(x) is defined as: Let be a positive integer and let The derivative of f(x) is: ### 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 be the probability that a head shows up in a toss, and let . The probability that there is head in the sequence of toss is: We know that sum of all the probability must equal to 1. In order to show this, we can use Binomial Theorem. We have: Please also check another article Gaussian Samples and N-gram language models ,Bayesian model, Monte Carlo for statistics knowledge.

# Hiring Data Scientist / Engineer

We are looking for Data Scientist and Engineer. 