In the binomial distribution, there are a number of probabilities, like:

- the trial number is indeterminately large, e., n -> ∞
- porq is minor, then the controlling form of the binomial distribution is known as ‘normal distribution’. It is thus a continuous distribution and the probability density function is given by,

f(x) = 1/ σ √2π. e^{-(x-µ)2/2σ2}, -∞<x<∞, -∞ <µ<∞

= 1/ σ √2π. exp [ – (x-µ)^{2}/2σ^{2}]

It is thus represented by N (11, cr2) where 1.1. = Mean and cr2 = Variance.

Thisarrangement was established by Carl Friedrich Gauss, which is also known as ‘Gaussian Distribution’.

**Properties**

(i) The normal curve is bell shaped and symmetrical about the .line x = µ.

(ii) Median: Let M be the median, then

^{M}ʃ_{-∞}f(x) dx = 1/ 2

= 1/ σ √2π ^{M}ʃ_{-∞}exp. [-(x-µ)^{2}/2σ^{2}]dx = 1/ 2

= 1/ σ √2π ^{M}ʃ_{-∞}exp. [-(x-µ)^{2}/2σ^{2}]dx + 1/ σ √2π ^{M}ʃ_{-∞}exp. [-(x-µ)^{2}/2σ^{2}]dx = ½

In the first integral we consider z= x-µ/σ, then what we obtain is

1/√2π ^{∞}ʃ_{0 }exp. [-z^{2}/2] dz = ½

So therefore, 1/ 2 + 1/ σ √2π ^{M}ʃ_{µ }exp. [-(x-µ)^{2}/2σ^{2}]dx = 1/ 2

1/ σ √2π ^{M}ʃ_{µ }exp. [-(x-µ)^{2}/2σ^{2}]dx =0

µ = M

(iii) Mode: It is the value of x for which f(x) is maximum i.e., f’ (x) = 0 and f” (x) < 0.

Therefore, we obtain, mode = µ

Note. Mean, Median and Mode might match.

(iv) Mean-deviation (M.D.)

**Links of Previous Main Topic:-**

- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution

**Links of Next Statistics Topics:-**

- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory