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Chapter 5. Continuous Probability Distributions Section 5.6: Normal Distributions

Jiaping Wang Department of Mathematical Science 03/27/2013, Wednesday

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Outline Probability Density Function Mean and Variance

More Examples Homework #9

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Part 1. Probability Density Function

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Probability Density Function In general, the normal density function is given by

𝑓 𝑥 =

1 exp 𝜎 2𝜋

−

𝑥−𝜇 2𝜎2

2

, −∞ < 𝑥 < ∞, where the

parameters μ and σ are constants (σ >0) that determines the shape of the curve.

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Standard Normal Distribution Let Z=(X-μ)/σ, then Z has a standard normal distribution 1

𝑧2 𝑓 𝑧 = exp − , −∞ < 𝑧 < ∞ 2 2𝜋

It has mean zero and variance 1, that is, E(Z)=0, V(Z)=1.

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Part 2. Mean and Variance

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Mean and Variance ∞

∞

𝒛𝟐 𝑬 𝒁 = 𝒛𝒇 𝒛 𝒅𝒙 = 𝒆𝒙𝒑 − 𝒅𝒛 𝟐 𝟐𝝅 −∞ −∞ ∞ 𝟏 𝒛𝟐 = 𝒛 ∙ 𝒆𝒙𝒑 − 𝒅𝒛 = 𝟎. −∞ 𝟐𝝅

𝐄 𝐙𝟐

∞

= −∞

𝐳𝟐

𝒛𝟐 𝟏 𝐞𝐱𝐩 − 𝐝𝐳 = 𝟐 𝟐𝝅 𝟐𝝅

𝒛

𝟐

∞ 𝟎

/ 𝒖𝟏 𝟐𝐞𝐱𝐩

𝒖 𝟏 𝟑 − 𝐝𝐮 = Γ 𝟐 𝟐𝝅 𝟐

𝟐

𝟑/𝟐

= 𝟏.

Then we have V(X)=E(X2)-E2(X)=1. As Z=(X-μ)/σX=Zσ+μE(X)=μ, V(X)=σ2.

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Calculating Normal Probabilities 𝑃 𝑧1 < 𝑍 < 𝑧2 = 𝑧2 1 𝑧2 exp − 𝑑𝑧= 𝑧1 2𝜋

for z1