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# Binomial Distribution

Binomial distribution is related to the Probability Theory in mathematics. Binomial distribution is discrete Probability distribution and frequently used to model the number of successes in a sample of size ‘n’ that is made by replacement from a population of size ‘N’.
Binomial Distribution is also named as Bernoulli trial that has following properties-
The experiment has ‘n’ number of repeated trials.
The trials can have two possible outcomes success or failure.
The trials are independent means the occurrence of one trial does not effects the occurrence of another trial.
Some Notations are there to understand binomial distribution-
‘n’ is the number of successes that results from an experiment.
‘N’ is the number of trials.
‘P’ is the probability of success on an individual trial and ‘q’ is the number of failure on an individual trial.
Mathematically,
Binomial distribution formula Pp (n | N) is given as-
Pp (n | N) = Ncn pn qN-n = N! / n! (N – n)! pn (1 – p)N – n
Where  Ncn is the binomial coefficient.
The Mean of the distribution ‘μ’ is given as ‘Np’.
The variance (ÍŤσ2) is Np (1 – p).
The Standard Deviation (σ) is √ (Np (1 - p)).
The probability of getting more successes than observed in a binomial distribution is denoted as-
P = Nk=n+1 Nck pk (1 – p)N-k,
P = IP (n + 1, N – n),
Where,
Ip (a,b) = B(x; a,b) / B (a,b),
Here the term B (a,b) is known as beta function and B(x; a,b) is the incomplete beta function.
The characteristic function for the binomial distribution is given as
Ő“ (t) = q + p eitN,
The moment generating function ‘M’ for the distribution is
M (t) = et n,
M (t) = n = 0N Ncn pn qN-n,
M (t) = n = 0N Ncn (p etn (1 – p)N – n,
M (t) = [ p et + (1 - p) ]N,
The differentiation of the moment generating function ‘M’ is
M' (t) = N [p et + (1 – p)] N-1 (p et).
The second derivative of the moment generating function ‘M’ is
M” (t) = N (N – 1) [p et + (1 – p)] N-2 (p et)2 + N [p et + (1 – p)] N-1 (p et),
Thus the mean can be calculated as
μ = M' (0),
μ = N (p + 1 – p)p,
μ = N p,
μ'1 = μ = Np,
μ'2 = Np (1 – p + Np),
μ'3 = Np (1 – 3p + 3Np + 2p2 – 3Np2 + N2p2),
μ'4 = Np (1 – 7p + 7Np + 12p2 – 18Np2 + 6N2p2 – 6p3 + 11 Np3 – 6N2p3 + N3p3),
Thus the moments about the mean can be written as-
μ2 = Np (1 – p) = Npq,
μ3 = Np (1 – p) (1 – 2p),
μ4 = Np (1 – p) [3p2 (2 – N) + 3p (N – 2) + 1],
You can understand this concept in detail with the help of binomial distribution examples.

## Binomial Distribution Problems

Binomial distribution problems are based on Bernoulli Trials which has three important possible conditions given as follows:
1.      We will get at least one consequence of two possible outcomes in each sample that are called as "success" or "failure".
2.      From one sample to next, the Probability of success will remain same.

## Mean of Binomial Distribution

The binomial Random Variable Y is the number of successes in the n repeated trials in the performed experiments. The Probability distribution of a binomial random variable Y is known as Binomial Distribution. The binomial distribution in the discrete Probability Distribution is denoted as P ( n | N ) for obtaining n number of successes out of N Bernoulli Tr...Read More

## Bernoulli Trials

When outcomes of an experiment are random, like sometimes outcome of that experiment is success or sometime outcome of that experiment is failure, then these trials are known as Bernoulli trials. For understanding the Bernoulli trials practically, we assume a collection of random variables as ‘Xj’ and the value of Random Variable varies with outcome like Xj = 1, if the

## Variance of Binomial Distribution

Binomial Distribution is related to Probability Theory in mathematics. This is the discrete Probability distribution of the number of successes in a sequence of ‘n’ independent experiments in which every event in the Sample Space has a probability of success as ‘p’. If the number ‘n’ in ‘n’ independent experiments makes 1 i.e. (n = 1), then this Binomia...Read More