Binomial Distribution part 1 deals with introduction & the derivation of pdf of B D under the syllabus of complementary statistics for BSc Mathematics, Physics & Computer Science.

1. Welcome
Suchithra’s Statistics Classes
Suchithra's Statistics Classes -- Binomial Distribution

2. Binomial Distribution – B.D
Part – 1
Intoduction
(Based on complementary Statistics of
Bsc , University of Calicut)
Suchithra's Statistics Classes -- Binomial Distribution

3. Suchithra's Statistics Classes -- Binomial Distribution
Suppose a sample of ‘n’ individuals, drawn from a given
population is divided into two groups according as they
possess a certain attribute or do not possess that
attribute. Such a division is called dichotomy.
For example the division may be into male and female,
blind and not blind, literate and not literate, etc.
If ‘x’ individuals possess the attribute under consideration;
‘x’ is an integer which may take any value from 0 to n,
where n is the number of trials (inclusive).

4. Suchithra's Statistics Classes -- Binomial Distribution
Let ‘p’ be the constant probability that an individual
selected at random will have this attribute (Success) from
the population.
Then ‘(1-p)’ = q is the probability of not possessing the
attribute (failure). i.e. p+q = 1 .
If we are repeating the trial n times, then that r.v is said to
follow a binomial distribution.

5. Suchithra's Statistics Classes -- Binomial Distribution
Binomial distribution is the sum of a series of multiple
independent and identically distributed Bernoulli trials.
The binomial distribution is often used in medical
science statistics as a building block for models for
dichotomous outcome variables, like whether a cure or
not cure a particular disease, whether an individual will
die within a specified period of time, etc.

6. Suchithra's Statistics Classes -- Binomial
Distribution
The calculation of pmf/pdf of binomial distribution can be
illustrated as follows:
Let we have n trials. ‘x’ be the number of success with
constant probability ‘p’. Then we have (n-x) failures with
probability (1-p).
One of the arrangement of these x success & (n-x)
failures trails be
with probability px (1-p)(n-x)
The number of arrangements of x success & (n-x) failures
is nC
x Required probability is nCx px (1-p)(n-x)

7. Suchithra's Statistics Classes -- Binomial Distribution
i.e. The probabilities of the Binomial distribution is
calculated by multiplying the probability of success ‘p’
raised to the power of the number of successes ‘x’ and the
probability of failure ‘1-p’= q raised to the power of the
difference between the number of successes and the
number of trials (n-x). ie. px qn-x
Then, multiply the product by the combination between
the number of trials and the number of successes nCx
P.S. nCx
nᴉ =n(n-1)........3.2.1

8. Suchithra's Statistics Classes -- Binomial Distribution
Definition
A random variable X is defined to have a binomial
distribution if the probability density function of X is given
by
f(x;n,p) = nCx px q n-x , for x = 0,1,2,.....,n
p+q =1,
= 0 , elsewhere.
where n & p are the parameters of B.D.
X b(x;n,p)

9. Suchithra's Statistics Classes -- Binomial Distribution
A distribution is called a Binomial Distribution
because the probabilities of the r.v X takes the value
as follows:
which are the successive terms of the binomial
expansion (q + p)n .Then X is called a binomial
variate.
Any r.v follows binomial distribution is called a
binomial variate.
X 0 1 2 ....... x ...... n
p(X =x) qn (nC1)qn-1p (nC2)qn-2p2 (nCx)qn-xpx pn

10. Suchithra's Statistics Classes -- Binomial Distribution
• Binomial distribution is a probability distribution that summarizes
the likelihood that a value will take one of two independent
values under a given set of parameters or assumptions.
• The underlying assumptions of the binomial distribution are that
there is only one outcome for each trial, that each trial has the
same probability of success, and that each trial is mutually
exclusive or independent of each other.
• Binomial distribution is a common discrete distribution used in
statistics, as opposed to a continuous distribution, such as the
normal distribution.

11. Things to know from this class
Define B.D
How to derive the pdf of B.D ?
What are the assumption of B.D?
What is a Binomial variate?
When we can assure that a r.v follows B.D.
Suchithra's Statistics Classes -- Binomial
Distribution

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Suchithra’s Statistics classes
Suchithra's Statistics Classes – Binomial Distribution