In the binomial distribution, there are a number of probabilities, like:
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’.
(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.)
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