5.2 Normal Distribution


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 xµ.
· Its curve has a maximum value at xµ.
· The area enclosed by the normal curve and the x-axis is 1.

2.  The notation of being normally distributed with a mean, µ and a variance, σ2 is X ~ (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 = x μ σ
where,
= standard score or z - score
= value of a normal random variable
µ = mean of a normal distribution
σ = standard deviation of a normal distribution

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