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 wide range of applications.
Theorem
Let ,
be real numbers (or complex, or polynomial). For any positive integer
, we have:
where,
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 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 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 probability model. Let be the probability that a head show up in a toss, and let
. The probability that there is
head in the sequence of
toss is:
Please also check another article Gaussian Samples and N-gram language models ,Bayesian model , Monte Carlo for statistics knowledges.