Normal Distribution

4.4.2 Normal Distribution

Normal Distribution has the following properties:

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Symmetrical and a bell shaped appearance.
The population mean and median are equal.
An infinite range, -∞ < x < ∞
The approximate probability for certain ranges of X-values:
P(μ - 1σ < X < μ + 1σ) ≈ 68%

P(μ - 2σ < X < μ + 2σ) ≈ 95%

P(μ - 3σ < X < μ + 3σ) ≈ 99.7%
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Normal Distribution:

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1 -(x-μ)2 f (x) = √-----e 2σ2 2πσ

-∞ < x < ∞
N(μ,σ²) is used to denote the distribution
E(X) = μ
V (X) = σ²
If μ = 0 and σ² = 1, this is called a Standard Normal and denoted Z
- All Normally distributed random variables can be converted into a Standard Normal
To determine probabilities related to the Normal distribution the Standard Normal distribution is used

4.4.2.1 Standard Normal Distribution:

The following formula is used to transform a Normally distributed random variable, X, into a Standard Normally distributed random variable,

Z = X----μ- σ

Note: if z is known we can solve for x:

x = μ + zσ

1 -z2 f(z) = √----e 2 2π

-∞ < z < ∞
Has the same properties as the Normal distribution and
- N(0, 1) is used to denote the distribution
- E(Z) = μ = 0
- V (Z) = σ² = 1
The approximate probability for certain ranges of Z-values:
P(-1 < Z < 1) ≈ 68%

P(-2 < Z < 2) ≈ 95%

P(-3 < Z < 3) ≈ 99.7%
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To find probabilities of any z-value refer to the Table ?? or computer software such as Microsoft Excel may be used

P(Z = c) = 0, where c is a constant.
P(Z < -c) = P(Z > c)
P(Z > c) = 1 - P(Z < c)
P(Z < -c) = 1 - P(Z < c)

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The Figures ??,??,??,??,?? illustrate the important probabilities associated with the normal distribution and calculating approximate probabilities using the information on the Normal distribution given in this subsection.

Figure 4.2: The Normal curve, f(z), and again with the area underneath shaded, representing P(-∞ < Z < ∞).

Figure 4.3: The area underneath the normal curve shaded from -1 to 1, and -2 to 2, illustrating P(-1 < Z < 1) ≈ 68% and P(-2 < Z < 2) ≈ 95% respectively.

Figure 4.4: The area underneath the normal curve shaded from -∞ to 0, and 0 to ∞, illustrating P(-∞ < Z < 0) = 50% and P(0 < Z < ∞) = 50% respectively.

Figure 4.5: The area underneath the normal curve shaded from -1 to 0, and 0 to 1, illustrating P(-1 < Z < 0) ≈ 34% and P(0 < Z < 1) ≈ 34% respectively. Resulting from symmetry.

Figure 4.6: The area underneath the normal curve shaded from -∞ to -1, and 1 to ∞, illustrating P(-∞ < Z < -1) ≈ 16% and P(1 < Z < ∞) ≈ 16% respectively. Resulting from symmetry.

4.4.2.2 The Distribution of the Average of I.I.D. Normally Distributed Random Variables

Let X₁,X₂,…,X_n be n i.i.d. N(μ,σ²) random variables. The expectation of the sample mean, , of n i.i.d. Normally distributed random variables is μ and the variance is σ2
n . Recall equations ?? and ??. The situation where the random variables are Normally distributed is a special case in that ¯
X is also Normally distributed and

σ2 ¯X ~ N (μ,---). n

In addition,

¯X - μ Z = --√---, σ∕ n

where Z ~ N(0, 1).

Examples

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The height of a randomly selected person is often assumed to be Normally distributed
The weight of a randomly selected person is often assumed to be Normally distributed
The pulse rate of a randomly selected person is often assumed to be Normally distributed

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