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 P_p (n | N) is given as-
P_p (n | N) = ^Nc_n pⁿ q^N-n = N! / n! (N – n)! pⁿ (1 – p)^{N – n}
Where  ^Nc_n is the binomial coefficient.
The Mean of the distribution ‘μ’ is given as ‘Np’.
The variance (͍σ²) 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 = _N∑^{k=n+1 N}c_k p^k (1 – p)^N-k,
P = I_P (n + 1, N – n),
Where,
I_p (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
Փ (t) = q + p e^itN,
The moment generating function ‘M’ for the distribution is
M (t) = e^{t n},
M (t) = _{n = 0}∑^N Nc_n pⁿ q^N-n,
M (t) = _{n = 0}∑^N Nc_n (p e^t) ⁿ (1 – p)^{N – n},
M (t) = [ p e^t + (1 - p) ]^N,
The differentiation of the moment generating function ‘M’ is
M' (t) = N [p e^t + (1 – p)] ^N-1 (p e^t).
The second derivative of the moment generating function ‘M’ is
M” (t) = N (N – 1) [p e^t + (1 – p)] ^N-2 (p e^t)² + N [p e^t + (1 – p)] ^N-1 (p e^t),
Thus the mean can be calculated as
μ = M' (0),
μ = N (p + 1 – p)p,
μ = N p,
The moments about zero are
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 + 6N²p^{2 –} 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.