Normal Random Variable
A continuous random variable, X, follows a normal distribution with mean, μ, and variance, σ², and is denoted by N(μ, σ²). It also must be noted:
1. The variable can take any value: (−∞, +∞)
2.The density function is the expression in terms of the mathematical equation of the Gaussian curve:
The Normal Curve
Domain: (−∞, +∞).
It is symmetric about the mean μ.
It has a maximum in the mean μ.
It's increasing to the mean μ and descending from it.
At points μ − σ and μ + σ it has inflection points.
The horizontal axis is an asymptote of the curve.
The area determined by the function and the x-axis is equal to 1.
Since it is symmetrical about the axis x = µ, it leaves equal areas on either side of 0.5.
The probability is equivalent to the area under the curve.
p(μ − σ < X ≤ μ + σ) = 0.6826 = 68.26 %
p(μ − 2σ < X ≤ μ + 2σ) = 0.954 = 95.4 %
p(μ − 3σ < X ≤ μ + 3σ) = 0.997 = 99.7 %