# PROPERTIES OF BINOMIAL DISTRIBUTION

## About "Properties of binomial distribution"

Properties of binomial distribution :

Students who would like to learn binomial distribution must be aware of the properties of binomial distribution.

Because, without knowing the properties, always it is difficult to solve probability problems using binomial distribution.

So, let us come to know the properties of binomial distribution.

## Properties of binomial distribution

1.  Binomial distribution is applicable when the trials are independent and each trial has just two outcomes success and failure.

It is applied in coin tossing experiments, sampling inspection plan, genetic experiments and so on.

2. Binomial distribution is known as bi-parametric distribution as it is characterized by two parameters n and p.

This means that if the values of n and p are known, then the distribution is known completely.

3.  The mean of the binomial distribution is given by

μ  =  np

4. Depending on the values of the two parameters, binomial distribution may be uni-modal or bi-modal.

To know the mode of binomial distribution, first we have to find the value of (n+1)p.

(n+1)p is a non integer --------> Uni-modal

Here, the mode  =  the largest integer contained in  (n+1)p

(n+1)p is a integer --------> Bi-modal

Here, the mode  =  (n+1|)p, (n+1)p - 1

5.  The variance of the binomial distribution is given by

σ²  =  npq

6.  Since p and q are numerically less than or equal to 1,

npq < np

That is, variance of a binomial variable is always less than its mean.

7.  Variance of binomial variable X attains its maximum value at p = q = 0.5 and this maximum value is n/4.

8. Additive property of binomial distribution.

Let X and Y be the two independent binomial variables.

X is having the parameters n₁ and p

and

Y is having the parameters n₂ and p.

Then (X+Y) will also be a binomial variable with the parameters (n₁ + n₂) and p

## Properties of binomial distribution - Practice problems

Problem 1 :

Find the binomial distribution for which mean and standard deviation are 6 and 4 respectively.

Solution :

Mean  =  6 ----> np  =  6 ------(1)

Standard deviation  =  2

Then, variance  =  4 ----> npq  =  4 ------(2)

Dividing (2) by (1), we get q  =  2/3.

And we know that p  =  1 - q

Thus, we get  p  =  1/3

Plugging p  =  1/3 in (1), we get

n x 1/3  =  6

n  =  18

Now, we know the values of "n", "p" and "q".

Therefore, the required binomial distribution is given by

Problem 2 :

What is the mode of the distribution for which mean and variance are 10 and 5 respectively.

Solution :

Mean  =  10 ----> np  =  10 ------(1)

Variance  =  5 ----> npq  =  5 ------(2)

Dividing (2) by (1), we get q  =  1/2.

And we know that p  =  1 - q

Thus, we get  p  =  1/2

Plugging p  =  1/2 in (1), we get

n x 1/2  =  10

n  =  20

To know the mode of a binomial distribution, first we have to know the value of (n+1)p.

(n+1)p  =  (20+1) x 1/2  =  10.5

Since the value of (n+1)p is a non integer, the given binomial distribution is uni-modal.

Hence mode  =  Largest integer contained in (n+1)p

=  Largest integer contained in (20+1) × 1/2

=  Largest integer contained in 10.50

=  10

After having gone through the stuff given above, we hope that the students would have understood "Binomial distribution properties".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6