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There is one type of continuous random variable whose pdf has a bell shape. It is called the normal random variable and it's pdf is called the normal distribution. It has two parameters: μ and σ with σ > 0. It is written as:
| f(x) = |
1 √2πσ |
* e(-1/2)((X - μ)/σ)2 |
and also as:
N( μ, &sigma2 )
When μ = 0 and σ = 1, its graph appears as follows:
This graph also represents the standard normal distribution.
The normal random variable is usually denoted with a Z and it corresponds to:
| Z = |
(X - μ) σ |
When μ = 0 and σ = 1, the normal distribution can be written as follows:
| fz(z) = |
1 √2π |
* e-z2/2 |
The way probability is measured using the bell curve is by finding the area underneath the curve. For example, to find a probability where the area is between z = 0 and z = 1, the following formula is used:
| P( a ≤ Z ≤ b ) = ∫ |
|
1 √2π |
e-z2/2dz |
| P( 0 ≤ Z ≤ 1 ) = ∫ |
|
1 √2π |
e-z2/2dz |
| = .34134 |
Between 0 and 1 is 34 percent of the area of the bell curve.
An area represented by two Z values has a 34 percent probability if it falls between 0 and 1.
The area between -1 and 1 corresponds to 1 standard deviation from the mean. Since the bell curve is symmetric about the mean, the area between -1 and 0 is also equal to .34134. Therefore, the area corresponding to -1 standard deviation to +1 standard deviation is .68268 or 68%.
As a result, the Z axis is measured in terms of standard deviations except that the standard deviations also include the area on the other side of the mean.
The areas under the curve for corresponding to the first three standard deviations are as follows:
σ = 1 : 68.3%
σ = 2 : 95.4%
σ = 3 : 99.7%
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