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LEC16 Normal Dist

LEC16 Normal Dist - The Normal Distribution A continuous...

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The Normal Distribution A continuous normal r.v. X has probability density function ( 29 f x e x x ( ) = - ∞ < < ∞ - - 1 2 2 2 2 σ π μ σ parameters: - < μ < and σ >0. E(X)= μ and V(X)= σ 2 Normal Distribution f(x) x
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When Does the Normal Distribution Arise Measurement is subject to sources of error that are many sources of variation independent sources each source contributes a small variation Example: dimension of a machined part is subject to variations in temperature humidity vibrations cutting angle cutting tool wear bearing wear rotational speed mounting and fixturing raw materials 2
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Learn the Probabilities of the Normal Distribution TRUE FOR ANY μ AND σ μ 68% μ-σ μ+σ P X ( ) . μ σ μ σ - < < + = 6827 μ μ+2σ μ-2σ 95% P X ( ) . μ σ μ σ - < < + = 2 2 9545 μ μ-3σ 99% μ+3σ P X ( ) . μ σ μ σ - < < + = 3 3 9973 3
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The Normal Distribution is Symmetrical Find the probability X is greater than μ +2 σ . μ μ+2σ μ-2σ .025 .95 .025 P(X> μ +2 σ ) = .025 Find the probability X is less than μ +2 σ . μ μ+2σ μ-2σ .025 .95 .025 P(X< μ +2 σ ) = .025+.95 = .975 4
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The Standard Normal The standard normal distribution has μ =0 and σ =1.
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