**5.2 Normal Distribution**

(A) Continuous Random Variable

(A) Continuous Random Variable

Continuous random variable is a variable that can take any infinite value in a certain range.

(B) Normal Distribution

(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, σ

^{2}is

*X*~

*N*(µ, σ

^{2}).

(C) Standard Normal Distribution

(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, σ

^{2}= 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

(E) Converting a Normal Distribution to Standard Normal Distribution

A normal distribution can be converted to the standard normal distribution using the following formula:

$$\overline{)Z=\frac{x-\mu}{\sigma}}$$

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