May 25

# Learn Functions and Binomial Distribution in Concise Form

In Mathematics, a topic like functions is one of the most important topics as this topic serves as a basic building for whole calculus (differential and integral). Therefore, you must know the important functions such as Linear, Constant, Composite Functions and others for their use in other important mathematics topics asked in various defence and engineering entrance exams.

Another concept that is used in high-weightage topics like statistics and probability in various competitive exams is the binomial distribution.

On that note, let’s discuss ‘functions’ and ‘binomial distribution’ separately to help you get a hang of these concepts from definition to their application with examples.

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## What is a Function?

A function is a connection that stipulates that each input should have only one output (or) it is a special form of relation (a set of ordered pairs) that follows a rule, i.e., each X-value should be coupled with only one y-value.

Let’s have a look at how the Domain and Range of a function are defined.

Domain: It is a collection of the first values of the ordered pair (Set of all input (x) values).

Range: It is a collection of the second values of the ordered pair (Set of all output (y) values).

Example:

In the relation, {(-1, 3), {7, 5), (6, -5), (-2, 3)},

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The domain is {-1, 7, 6, -2} while the range is {-5, 3, 5}.

### Types of Functions

Function types can be defined using the following relationships:

#### One to one function or Injective function

f: P is a function. If each element of P has a separate element of Q, Q is said to be one-to-one. The domain is {-2, 4, 6} while the range is {-5, 3, 5}.

#### Many to one function

This function maps two or more P elements to the same Q element.

#### Onto Function or Surjective function

This function has a pre-image in set P for each element of set Q.

#### One-one correspondence

Each element of P corresponds to a discrete element of Q, and each element of Q corresponds to a pre-image in P.

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#### Special Functions in Algebra

The following are some of the most important functions:

Identity Function

Constant Function

Inverse Functions

Absolute Value Function

Linear Function

Composite function

## Binomial Distribution

The binomial distribution is a discrete probability distribution in probability theory and statistics that gives only two possible outcomes in an experiment: success or failure.

For example, if we flip a coin, there are only two possible outcomes: heads or tails, and if we take any test, there are only two possible outcomes: pass or fail. This is also known as a binomial probability distribution.

For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution.

Other real-life examples, where the concept is used can be a yes/no survey to check the popularity of a channel, votes collected by a candidate in an election, and more.

### Negative Binomial Distribution

The number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified number of failures occurs in probability theory and statistics. It is termed the negative binomial distribution and is denoted by ‘r’.

For example, if we throw a dice and count every 1 as a failure and every non-1 as a success. If 1 appears for the third time, i.e., r = three failures, the probability distribution of the number of non-1s arriving is the negative binomial distribution.

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### Binomial Distribution Formula

For each random variable X, the binomial distribution formula is given by;

P(x:n,p) = nCx px (1-p)n-x

Where,

n = the number of experiments

x = 0, 1, 2, 3, 4, …

p = Probability of Success in a particular experiment

1 – p = Probability of Failure in a particular experiment

For n-Bernoulli trials, The formula is given as:

P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

### Binomial Distribution Mean and Variance

The formulas reflect the mean, variance, and standard deviation for a binomial distribution with a certain number of successes.

Mean, μ = np

Variance, σ2 = npq

Standard Deviation σ= √(npq)

Where p = probability of success

q = probability of failure =1-p