5.2 Normal Distribution - SPM Additional Mathematics

5.2 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 = \frac{x - μ}{σ}

where,

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