Binomail distribution 23 jan 21

Binomial & Poisson Distribution
(For M.B.A. I Semester)
by Prof. Nitin Karoulia
9033099006

Combination (Definition). A combination of n different
objects taken r at a time, denoted by 𝒏𝑪𝒓 or n ,is a
selection of only r objects out of the
r
n objects, without any regards to the order of arrangement.
Theorem 12.6. The number of different combinations of n
different objects taken r at a time, without repetition, is
𝒏𝑪𝒓= n = n ! ; r < n
r r ! (n – r) !

Binomial Distribution
Binomial distribution was discovered by Swiss Mathematician James
Bernoulli, so this distribution is called as ‘Bernoulli Distribution’ also. This
is a discrete frequency distribution.
Assumptions of a Binomial Distribution
1. The random experiment is performed repeatedly with a fixed and finite
number of trials. Number of trials is denoted by ‘n’.
2. There are two mutually exclusive possible outcomes on each trial, which
are known as ‘success’ and ‘failure’. Success is denoted by p whereas
failure is denoted by q, and p+ q = 1, or q = 1 - p.
3. The outcomes of any given trial does not affect the outcomes on
subsequent trials, means the trials are independent.
4. The probability of success and failure (p & q) remains constant from trial
to trial. If in any distribution the p & q does not remain constant, that
distribution cannot be a binomial distribution. For example, the
probability of getting head (or tail) must remain the same in each toss,
i.e.1/2. Similarly the probability of drawing 4 balls from a bag containing
6 red and 10 white balls does not change in successive draws with
replacement, hence it will be called as binomial distribution. But, in
contrary to this, if balls are not replaced after each trial, then it will not
be a binomial distribution.

Binomial distribution is the expansion of (𝑝 + 𝑞)𝑛, from which the probability
of number of success (r) in descending order (…, 4, 3, 2, 1, 0) can be
determined. For determining the probability in ascending order we use
the expansion of (𝑞 + 𝑝)𝑛 . For example if n = 5, p = probability of
success (Heads), q = probability of failure (trials) then the expansion of
(𝑝 + 𝑞)5 & (𝑞 + 𝑝)5 may be given as follows :

Expected Frequencies of Binomial Distribution
For determining the expected frequencies, the probability of each item is
multiplied by total number of trials (N)
N(p + q)n or N(q + p)n
N = Total number of trials.
n = Number of independent events (coins, dices, etc.)
Constants of Binomial Distribution
Binomial Distribution is a special type of distribution, the constant values of it
i.e. Mean (X) and Standard Deviation (s) Can be computed by using the
following formulae_ :
X = np
σ = √npq

Assumptions of Binomial Distribution
(1) Trials are repeated.
(2) All trials should be independent.
(3) The number of trial (n) should be fixed.
(4) There are two mutually exclusive outcomes in each trial,
success and failure.

Mean = np (7.2)
Variance = npq (7.3)
Standard Deviation = √ npq (7.4)
It may be noted that the variance (npq) is always less than the
mean (np) because q which multiplies np, is always less than
1, and, therefore, reduces the value of np. Thus, in the
Binomial Distribution, variance can never exceed the mean.
For example, if n = 10, p = 0.3 and q = 1 – p = 0.7
Then, np = 10 x 0.3 = 3, and npq = 10 x 0.3 x 0.7 = 3 x 0.7 = 2.1
which is less than 3.

It may be noted that given values of n and p, one can find the
values of mean (= np), and variance(npq) , On the other hand,
given the values of mean and variance, one can find the
values of n and p.
For example, if mean = 3.0, and variance = 2.1, it implies that np
= 3.0 and npq = 2.1, and, therefore,
q = mean = npq = 3.0 = 0.7
variance np 2.1
and p = 1 – q = 1 – 0.7 = 0.3
Since mean = np = 3.0, and p = 0.3,
we have
n = np = 3.0 = 10
n 0.3
it implies that n = 10, p = 0.7 and q = 0.3

Distribution Function : While f(x) is known as the probability
function or probability distribution, the value of
x
∑ f (x)
x = 0
Denoted by F (x) is called the cumulative distribution function or simply
distribution function. It is similar to cumulative frequency i.e the
cumulative frequency up to a given point or value. It gives the probability
that the random variables takes any value from 0 (lowest value that the
variable can take) to a given value x. Thus, in Illustration 7.1,
F(0) = f(0) = ¼
F (1) = ∑ f (x)
= f(0) + f(1)
= ¼ + ½ = 3/4

Similarly
F (2) = ∑ f (x)
= f(0) + f(1) + f(2)
= ¼ + ½ + ¼ = 1
The shape of the above distribution function is depicted below.

Properties of Binomial Distribution
There are two important properties of any probability function including the
Binomial Distribution:
(i) f(x)≥ 0 for each and every value of x which it can take.
(ii) ∑f(x) = 1 summation being over all values of x which it can take
In addition , the other properties are
(i) Range of the variable x is from 0 to n, where n is the number of trials
(ii) Mean = np
(iii) Variance = npq, and is always less than mean
(iv) Shape of the distribution is symmetrical for p = ½ , skewed to the right
when p < ½, and skewed to the left when p > ½.

Obtaining Coefficients of the Binomial
For obtaining coefficients from the binomial expansion, the
following rules may be remembered. To find the terms of the
expansion of (q + p)
1. The first term is q
2. The second term is C1 q p
3. In each succeeding term the power of q is reduced by 1 and
the power of p is increased by 1.
4. The coefficient of any term is found by multiplying the
coefficient of the preceding term by the power of q in that
preceding term, and dividing the products so obtained by
one more than the power of p in that preceding term..

When we expend (q + p) , we get

Illustration 1 : Six coins are thrown simultaneously. Find the chance of
obtaining (i) no head, (ii) at least one head, (iii) exactly two heads, (iv) not
more than two heads; (v) more than 3 heads.
Solution :

Poisson Distribution
Poisson Distribution was introduced by S.D. Poisson as a distribution of rare
events i.e. the events whose probability of occurrence is very small but
the number of trials, which could lead to the occurrence of the event, are
very large. Some of such situations are :
• Number of air accidents in a day – there are thousands of flights in a day
but the probability of a plane meeting with an accident is very small. This
leads to very few accidents. Similar is the situation with car accidents or
accidents in factories, etc.
• Number of customers arriving at a bank’s counter every minute – there
are thousands of customers but the probability of a customer arriving at a
counter, in a minute, is very small. This leads to few customers reaching
the counter in a minute.
• Number of telephone calls coming to an office every minute – the office
can receive call from thousands of phones but the probability of receiving
a call from each of the phones is very small. This is the reason for receiving
very few calls in a time span of one minute

• Number of defects in pieces of clothes of a certain length, say one metre –
the cloth being of average quality- there are thousands of points on the
cloth where a weaving or some other defect could have occurred but in a
average quality cloth, the probability of a point being defective is very
small. This is evident from the fact that the number of defects on every
metre of cloth is very few. Similar is the situation about a roll of paper,
plastic, metal sheet, etc.
• Number of goals scored per match in a soccer tournament – During the 90
minute duration of a match, a goal could be scored any moment during
the match, but the probability of a goal being scored every moment is very
small. This is the reason for either no goal or few goals being scored in a
match.
Some other situations where Poisson distribution could be applied are given
in the Section 7.7.
Usually, Poisson distribution is applicable in situations that involve counting,
and the counts are rare.

Where n is very large but p is very small such that their product np is finite,
say m, the Binomial Distribution tends to a function referred to as Poisson
Distribution. It is defined as
f(x) = e m x = 0,1,2,3,………… ∞ (7.5)
x!
The conditions for the applicability of a Poisson distribution are:
• The variable (number of occurrences ) is a discrete variable.
• The occurrence are random
• The occurrence are rare.
Mean and Variance Just like Binomial distribution, the mean of Poisson
distribution is defined as

Shape of Poisson Distribution The shape of a Poisson distribution depends on
the value of m. Shapes of the Poisson distribution for some values of m
are shown below :
The distribution has no definite shape for different values of m. It gets more
and more symmetrical about the mean as ‘m’ increases.
Properties of Poisson Distribution
(i) Range of the variable is from 0 to ө
(ii) Mean and Variance are equal
(iii) Distribution gets symmetrical and tends to normal distribution,
described in the Section 7.5.2, as mean increases

Examples:
Some examples have been solved below to illustrate computation of probabilities
with the help of Poisson distribution.
Example 7.4
In a certain manufacturing process, 2% of the tools produced turn out to be
defective . Find the probability that in sample of 50 tools, at least 3 will be
defective.
Solution:
The situation described here is such that Binomial distribution is applicable with n
= 50 and p = 0.02. However, since n ≥ 30 and p ≤ 0.05, we can use the Poisson
distribution with mean m = np = 50 x 0.02 = 1 as an approximation to the
Binomial distribution. The variable x is the number of defectives in a sample of
50 tools.
We have to find out
P(x = at least 3 defectives)
= 1 – P (x = at most 2 defectives)
= 1 – P (x ≤ 2)
= 1 – P(x = 0) – P (x = 1) – P (x = 2)

Binomail distribution 23 jan 21

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Binomail distribution 23 jan 21