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)2/2σ2]
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
Mʃ-∞f(x) dx = 1/ 2
= 1/ σ √2π Mʃ-∞exp. [-(x-µ)2/2σ2]dx = 1/ 2
= 1/ σ √2π Mʃ-∞exp. [-(x-µ)2/2σ2]dx + 1/ σ √2π Mʃ-∞exp. [-(x-µ)2/2σ2]dx = ½
In the first integral we consider z= x-µ/σ, then what we obtain is
1/√2π ∞ʃ0 exp. [-z2/2] dz = ½
So therefore, 1/ 2 + 1/ σ √2π Mʃµ exp. [-(x-µ)2/2σ2]dx = 1/ 2
1/ σ √2π Mʃµ exp. [-(x-µ)2/2σ2]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