The document explains the binomial theorem, which facilitates the expansion of expressions like (a+b)^n, and discusses the binomial distribution representing trials with binary outcomes (success or failure). It also covers the poisson distribution, focusing on its application to predict the probability of events occurring within a specific interval, with notable applications in various fields such as call centers, website traffic, and banking. The historical origins of both distributions are briefly outlined, highlighting key contributors like Isaac Newton for the binomial theorem and Siméon-Denis Poisson for the poisson distribution.

1. PRESENTATION
Binomial Theorem and Distribution
Versus Poisson Distribution

2. The Binomial Theorem
The Binomial Theorem tells us how to expand expressions of the form (a+b)ⁿ,
For example, (x+y)⁷. The larger the power is, the harder it is to expand
expressions like this directly. But with the Binomial theorem, the process is
relatively fast!
The binomial theorem is used for the expansion of the algebraic for:
(p+q)n
= ∑ n
Cr
pr
qn-r
(p+q)n
= n
C0
p0
qn-0
+ n
C1
p1
qn – 1
+ n
C2
p2
qn – 2
+ n
C3
p3
qn – 3
+ ….. + n
Cn
pn
qn
- n

3. (p+q)n
= ∑ n
Cr
pr
qn-r
(p+q)n
= n
C0
p0
qn-0
+ n
C1
p1
qn – 1
+ n
C2
p2
qn – 2
+ n
C3
p3
qn – 3
+ ….. + n
Cn
pn
qn
- n
n → the number of repeated trials
r → the number of successful trials
p → probability of success
q → probability of failure
n
Cr
→ combination of n and r
∑→ sum of all
Combination is the number of ways to
choose a sample of x elements from a set
of n distinct objects where order doesn't
matter, and replacements are not allowed
Expanding the binomial theorem to power of ‘n’

4. What is Binomial Distribution?
Binomial distribution is a common probability distribution that is discrete and gives
only two possible results in an experiment, either Success or Failure.
Each of these experiment, also called Bernoulli trial shows that each trial has the
same chance of attaining one specific outcome.
Examples:
1) Tossing a coin 10 times, and counting the number of
face-ups is 50 percent, i.e p=0.5 and q=0.5
2) Counting the number of green-eyed people among 500
randomly chosen people (assuming that 5% of all people
have green eyes). (n=500, p=0.05)

5. Why we use Binomial theorem?
● The Binomial Theorem is used in advanced mathematics and calculating to
determine the roots of equations in higher powers.
● It's also used to prove a lot of important physics and math equations. E.g
Weather forecast services, architecture, and cost estimation in engineering
projects.
● Automatic distribution of IP addressing for computer networking.
● Economic forecasting all around the world the economist use to forecast the
growth of a country, a corporation and other
enterprises to understand how economy will
perform in near future.
● Used in ranking to determine scores and ranks.

6. History of Binomial
Theorem
Isaac Newton discovered about 1665 and
later stated, in 1676, without proof, the
general form of the theorem (for any real
number n), and a proof by John Colson
was published in 1736

7. A Poisson distribution is a discrete probability distribution. It gives the
probability of an event happening a certain number of times (k) within a given
interval of time or space. The Poisson distribution has only one parameter, λ
(lambda), which is the mean number of events
In Poisson distribution, the mean is represented as E(X) = λ. For a Poisson
Distribution, the mean and the variance are equal. It means that E(X) = V(X)
Where, V(X) is the variance.
Poisson Distribution

8. The concept of Poisson’s distribution is highly used by the call centres to
compute the number of employees required to be hired for a particular job. For
instance, if the number of calls attended per hour at a call centre is known to be 10,
then the Poisson formula can be used to calculate the probability of the organisation
receiving zero calls, one call, two calls, three calls, and any other integer number of
calls per hour, thereby allowing the managers to have a clear idea of the number of
calls required to be catered at different hours of the day and helps to form a proper
schedule to be followed by the employees accordingly.

9. The number of visitors visiting a website per hour can range from zero to
infinity. the Poisson probability distribution is most suited to calculate the probability
of occurrence of certain events. For instance, if the number of people visiting a
particular website is 50 per hour, then the probability that more or less than 50
people would visit the same website in the next hour can be calculated in advance
with the help of Poisson distribution. Once the probability of visitors about to visit a
particular website is known, the chances of website crash can be calculated. The
site engineer, therefore, tends to maintain the data uploading and downloading
speed at an adequate level, assigns an appropriate bandwidth that ensures
handling of a proper number of visitors, and varies website parameters such as
processing capacity accordingly so that website crashes can be avoided.

10. Poisson distribution is used by cell phone companies and wireless service providers
to improve their efficiency and customer satisfaction ratio. Provided that the history of
the number of network failures occurring in the locality in a particular time duration is well
known, the probability of a certain number of network failures occurring in future can be
determined easily with the help of Poisson distribution. This helps the broadcasting
organisations be prepared for the problems that might occur and draft the solution in
advance, so that the customers accessing their services don’t have to suffer the
inconvenience.

11. Poisson distribution finds its prime application in the banking sector. It is usually used
to determine the probability of customer bankruptcies that may occur in a given time.
For instance, if the bank records show that each month in a particular locality on
average four bankruptcies are being filed, then this information can be used to estimate
the probability of zero, one, two, or three bankruptcies may be filed in the following
month. This helps the bank managers estimate the amount of reserve cash that is
required to be handy in case a certain number of bankruptcies occur.
This has widespread applications, for example in analysing traffic flow, in fault
prediction on electric cables and in the prediction of randomly occurring accidents

12. The History of the Poisson Distribution
In1830 The French mathematician Siméon-Denis Poisson developed function to
describe the number of times a gambler would win a rarely won game of chance in a
large number of tries and named it Poisson distribution.
The Poisson is derived from the Binomial distribution, when one takes the limit of n, i.e.
the number of experiments, to infinity, while demanding that the expected value stays
the same (essentially reducing the probability to 1/n) it means that there is no two (1 or
0) events, but possibly infinite amount of events.
Thank you