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L13posted - NORMAL Distribution Most important distribution...

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NORMAL Distribution Most important distribution in statis- tics f ( x ) = 1 σ 2 π e - 1 2 x - μ σ 2 1
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E [ X ] = Var X = P ( X x ) = F ( x ) = Z x t = -∞ 1 σ 2 π e - 1 2 t - μ σ 2 dt No formula for this integral. Ta- bles exist C4 p568-9 for “stan- dard normal” Standard Normal μ = 0 , σ = 1 . Notation: Z for random variable, φ ( z ) pdf, Φ( z ) cdf. 2
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Using Standard Normal Table P ( Z 1 . 5) = P ( Z > 1) = P (1 < Z 1 . 5) = Finding percentiles of Z . Find 80 th percentile = z 0 . 80 of standard normal. Look in body of normal table for value nearest to 0.80. Read corresponding z . May need to interpolate. 3
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68-95-99.7% Rule for Z P ( - 1 < Z 1) = P ( - 2 < Z 2) = P ( - 3 < Z 3) = Note . Doesn’t matter whether use < or because normal is continuous. 4
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Standardization. Suppose want probabilities for normal X N ( μ = 10 , σ = 2) , say P ( X 13) . Write Z = X - μ σ Shown in text that Z is standard normal. Convert P ( X 13) into statement about Z . This is called standardization. Converts X into dimensionless quantities and pre- serves probabilities. 5
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P ( X 13) = = = = 0 . 9332 6
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General Standardization . Suppose X N ( μ, σ ) . P ( a < X b ) = P ( a - μ σ < Z <
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