Binomial Distribution Calculation

 Binomial Distribution is a discrete probability distribution. It was discovered by Swiss mathematician James Bernoulli(1654-1705). So it is also known as Bernoulli Distribution. This theorem was published eight years after the death of Jem Bernoulli. This theorem is applied when trials are conducted on mutually exclusive events with two options success(p) and failure(q).

The binomial distribution can be used with some assumptions:

a) Mutually Exclusive Outcomes.

b) Independent Trials.

c)Finite Number of Trials.

d)Probability of success is constant.

Properties of binomial distribution

1) It is discrete frequency distribution, Where the number of successes can't be a fraction.

2) The binomial distribution can be represented graphically by means of a line graph. The number of successes is represented on the x-axis and the expectations are represented on the y-axis.

3)If the probability of success in each trial is equal to the probability of failure(p=q) then the binomial distribution will be a symmetrical distribution.

4) If the probability of success is not equal to the probability of failure then the binomial distribution will be positively or negatively skewed.

5) The binomial distribution has two main parameters 'n' and 'p'.

6) If np(mean of binomial distribution) is a whole number, the distribution is unimodal distribution and the value of mean and mode are equal.

7) When the number of trials is large binomial distribution tends to be a normal distribution.

8) It is theoretical frequency distribution that is based on the binomial theorem of algebra.

Constants of Binomial Distribution

The constants of binomial distribution can be obtained by using some formulae:

Arithmetic Mean = mean x = np

[n = size of sample, P = Probability of success,q=1-p]

Standard Deviation =squre root(npq)

Variance = npq

Moment about mean:

mu1 = 0

mu2 = npq

mu3 = npq(q-p)

mu4 = 3*n^2*p^2*q^2

Skewness(Beta1):

Moment coefficient of skewness = +-squre root(B1)

where   B1 = mu3^2/mu^3

If  B1 = 0 then no skewness, if +ve then positively skewed if less than zero then negatively skewed.

Kurtosis(B2):

Moment coefficient  of kurtosis = gama2

where gama2=B2-3

   and  B2 = mu4/(mu2)^2

   if  B2 = 3 then the distribution will be Mesokurtic

   if  B2 >3 then Laptokuritic

   If  B2<3 then Platykuritic

Uses of Binomial Distribution

The binomial distribution is very useful in those fields where the outcomes can be classified only in the form of two attributes success and failure. And where the possibility of outcomes of any trial does not change and is independent of the result of previous trials. It can be used where:

a) Size of sample 

b) Neither p nor q is very small.

So it is very useful in coin experiments, dice throwing, manufacturing of items by a company, etc.

Fitting of Binomial Distribution

1)In respect of each term

2) In respect of all term 

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Normal Distribution

F-TEST

Option Greeks