1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 3 As moves farther and father away

1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 3 as

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1 2𝜎 2 𝜇+?−𝜇 2 = 1 2𝜋𝜎 ? 1 2𝜎 2 ? 2 ? 𝜇 − ? = 1 2𝜋𝜎 ? 1 2𝜎 2 𝜇−?−𝜇 2 = 1 2𝜋𝜎 ? 1 2𝜎 2 ? 2 = ?(𝜇 + ?) 3. As ? moves farther and father away from 𝜇 , ? ? → 0
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28 Shape of the Normal Distribution ) , ( 2 1 N ) , ( 2 2 N ) , ( 2 1 N ) , ( 2 2 N The mean is the central tendency of the distribution. It defines the location of the peak for normal distributions . Most values cluster around the mean. On a graph, changing the mean shifts the entire curve left or right on the 𝑿 -axis. The standard deviation is a measure of variability. It defines the width of the normal distribution . On a graph, changing the standard deviation either tightens or spreads out the width of the distribution along the 𝑿 -axis. Larger standard deviations produce distributions that are more spread out.
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29 Standard Normal Distribution The standard normal distribution 𝑁(0,1) is a normal probability distribution with 𝜇 = 0 and 𝜎 2 = 1 , [Special Case/Extremely Important] Probability density function (pdf) of 𝑁(0,1) : −∞ < ? < ∞ ? ? = 1 2𝜋𝜎 ? 1 2𝜎 2 𝑥−𝜇 2 = 1 2𝜋 ? 1 2 𝑥 2 ? ? = ?(−?) : The distribution is symmetric about 0 Mean = Median = Mode = 0
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30 Standard Normal Distribution Suppose that ? ~ 𝑁(𝜇, 𝜎 2 ) and we want to find Pr(? ≤ ? ≤ ?). ? ≤ ? ≤ ? ⇒ ? − 𝜇 ≤ ? − 𝜇 ≤ ? − 𝜇 ? − 𝜇 𝜎 ? − 𝜇 𝜎 ? − 𝜇 𝜎 ? − 𝜇 𝜎 ? − 𝜇 𝜎 ? − 𝜇 𝜎 The cdf of 𝑁(𝜇, 𝜎 2 ) cannot be obtained in closed form Pr ? ≤ ? ≤ ? = Pr ? − 𝜇 𝜎 ≤ ? ≤ ? − 𝜇 𝜎 ? − 𝜇 𝜎 ≤ ? ≤ ? − 𝜇 𝜎 ? = ? − 𝜇 𝜎 Centralize ?~𝑁(0,1) Shape of normal distribution preserves after translation (by −𝜇 ) and rescaling (by 1/𝜎 )
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31 Cumulative Distribution Function of 𝑵(?, ?) Notation for the cdf of 𝑁 0,1 : Φ ? = ? ? = Pr(? ≤ ?) Unable to compute the exact value of Φ ? by hand No closed form
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32 Cumulative Distribution Function of 𝑵(?, ?)
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33 Cumulative Distribution Function of 𝑵(?, ?)
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34 Cumulative Distribution Function of 𝑵(?, ?)
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35 Cumulative Distribution Function of 𝑵(?, ?)
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36 Cumulative Distribution Function of 𝑵(?, ?) Example: Please double-check the values below by yourself based on the normal table. Φ 0 = Pr ? ≤ 0 = 0.5 Φ 0.13 = Pr ? ≤ 0.13 = 0.5517 Symmetric around 0 Φ 1.96 = Pr ? ≤ 1.96 = 0.9750 Φ 3.90 = Pr ? ≤ 3.90 = 1.0000 Φ 1.645 = 0.9500 Between Φ(1.64) and Φ(1.65) How to obtain the cdf of a negative value of ? ?
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37 Cumulative Distribution Function of 𝑵(?, ?) Example: Please double-check the values below by yourself based on the normal table. Φ −0.5 Φ −1 Φ −1.96 Φ −2.58 Symmetry Properties Φ −? = Pr ? ≤ −? = Pr ? ≥ ? = 1 − Φ(?) = 0.3085 = 0.1587 = 0.025 0 = 0.0049 Example: Compute Pr(−1 ≤ ? ≤ 1.5) assuming ?~𝑁(0,1) Solutions: Pr −1 ≤ ? ≤ 1.5 = Pr ? ≤ 1.5 − Pr ? ≤ −1 = Pr ? ≤ 1.5 − Pr ? ≥ 1 = 0.7745
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