L13posted

L13posted - NORMAL Distribution Most important distribution...

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Unformatted text preview: NORMAL Distribution Most important distribution in statistics 1 1 −2 f (x) = √ e σ 2π x−µ 2 σ 1 E [X ] = Var X = P (X ≤ x) = F (x) 2 1 x − 1 t−µ = t=−∞ √ e 2 σ dt σ 2π No formula for this integral. Tables exist C4 p568-9 for “standard normal” Standard Normal µ = 0, σ = 1. Notation: Z for random variable, φ(z ) pdf, Φ(z ) cdf. 2 Using Standard Normal Table P (Z ≤ 1.5) = P (Z > 1) = P (1 < Z ≤ 1.5) = Finding percentiles of Z . Find 80th percentile = z0.80 of standard normal. Look in body of normal table for value nearest to 0.80. Read corresponding z . May need to interpolate. 3 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 Standardization. Suppose want probabilities for normal X ≈ N (µ = 10, σ = 2), say P (X ≤ 13). Write X −µ Z= 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 preserves probabilities. 5 P (X ≤ 13) = = = = 0.9332 fig_ 0 4 _ 1 5 6 General Standardization. Suppose X ≈ N (µ, σ ). a−µ a−µ <Z< ) P (a < X ≤ b ) = P ( σ σ Special Case. 68 − 95 − 99.7% Rule. Take a = µ ± σ or a = µ ± 2σ or a ± µ + 3σ. Then µ cancels in numerator and then σ cancels in numerator and denominator, reducing to the same probabilities as the standard normal 68 − 95 − 99.7% Rule. 7 68-95-99.7% Rule for X P (µ − σ < X ≤ µ + σ ) = P (µ − 2σ < X ≤ µ + 2 σ ) = P (µ − 3σ < X ≤ µ + 3 σ ) = fig_ 0 4 _ 1 2 8 Example 3.2-1, p 166. X ≈ N (µ = 75, σ 2 = 100). P (80 < X < 95) = = = = 0.9332 − 0.6915 = 0.2857. 9 Finding percentiles of X . xp = µ + σzp Find the 95th percentile of the normal X in previous example. x0.95 = = = 91.45 10 Time to assemble item is X ≈ N (µ = 15.8, σ = 2.4). Find P (X > 17). P (X > 17) 17 − 15.8 X − 15.8 > ) = P( 2. 4 2. 4 = P (Z > 0.5) = 1 − Φ(0.5) = 1 − 0.6915 = 0.3085. 11 ...
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This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.

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L13posted - NORMAL Distribution Most important distribution...

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