8.2 Normal Distribution
(A) Continuous Random Variable
Continuous random variable is a variable that can take any infinite value in a certain range.
(B) Normal Distribution
1. A continuous random variable, X, is normally distributed if the graph of its probability function has the following properties.
· Its curve has a bell shape and it is symmetrical at the line x = µ.
· Its curve has a maximum value at x = µ.
· The area enclosed by the normal curve and the x-axis is 1.
2. The notation of X being normally distributed with a mean, µ and a variance, σ2 is X ~ N (µ, σ2).
(C) Standard Normal Distribution
If a normal random variable, X, has a mean, µ = 0 and a standard deviation, σ = 1, then X follows a standard normal distribution, i.e. X ~ N (0, 1).
(D) Curve of a Standard Normal Distribution
1. The curve of a standard normal distribution has the following properties.
· Its curve is symmetrical at the vertical line that passes through the mean, µ = 0 and has a variance, σ2 = 1.
· Its curve has a maximum value at Z = 0.
· The area enclosed by the standard normal curve and the z-axis is 1.
(E) Converting a Normal Distribution to Standard Normal Distribution
A normal distribution can be converted to the standard normal distribution using the following formula:
Z = standard score or z - score
X = value of a normal random variable
µ = mean of a normal distribution
σ = standard deviation of a normal distribution