Class 4 - Normal Distribution

# Class 4 - Normal Distribution - For Continuous variables,...

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For Continuous variables, probabilities are the area under curve. For the standard normal distribution, probabilities can be the table in the book on A-42 and A-43. Also the normal DIS functions in Excelcan provide probabilities like using the norma Both use a value to find a probability. The t-distribution (A-44) with df = (infinity) is the same a Standard Normal Distribution. The INV functions in Excel oprate like the table in the book on where the probabilities are given at the top and the correspond values can be found in the table.

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r the found in T al table. as the A-44 ding
Standard Deviation = 1 Normal Parameters: Mean = 1 Standard Deviation = 0.5 X f(X) or f(Y) f(Z) -3 0 0 MIN -3 -2.93 0 0.01 MAX 3 -2.85 0 0.01 RANGE 6 -2.78 0 0.01 -2.7 0 0.01 -2.63 0 0.01 -2.55 0 0.02 -2.48 0 0.02 -2.4 0 0.02 -2.33 0 0.03 -2.25 0 0.03 -2.18 0 0.04 -2.1 0 0.04 -2.03 0 0.05 -1.95 0 0.06 -1.88 0 0.07 -1.8 0 0.08 -1.73 0 0.09 -1.65 0 0.1 -1.58 0 0.12 -1.5 0 0.13 -1.43 0 0.14 -1.35 0 0.16 -1.28 0 0.18 -1.2 0 0.19 -1.13 0 0.21 -1.05 0 0.23 -0.97 0 0.25 -0.9 0 0.27 -0.82 0 0.28 -0.75 0 0.3 -0.67 0 0.32 -0.6 0 0.33 -0.52 0.01 0.35 -0.45 0.01 0.36 -0.37 0.02 0.37 -0.3 0.03 0.38 -0.22 0.04 0.39 -0.15 0.06 0.39 -0.07 0.08 0.4 0 0.11 0.4 0.08 0.14 0.4 0.15 0.19 0.39 0.23 0.24 0.39 0.3 0.3 0.38 0.38 0.37 0.37 0.45 0.44 0.36 0.53 0.51 0.35 0.6 0.58 0.33 Standard Normal Mean = 0 The Sharpe text uses X to denote a variable in Chapter 21 but use Y to denote a normal variable that is not a standard normal. Probabilities are areas under a curve for continuous variables and can be found by integrating the function but the above formula can't be integrated in closed form so probabilities are determined by numerical methods and displayed in tables or calulated by functions.

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0.68 0.65 0.32 0.75 0.7 0.3 0.83 0.75 0.28 0.9 0.78 0.27 0.98 0.8 0.25 1.05 0.79 0.23 1.13 0.77 0.21 1.2 0.74 0.19 1.28 0.69 0.18 1.35 0.62 0.16 1.43 0.56 0.14 1.5 0.48 0.13 1.58 0.41 0.12 1.65 0.34 0.1 1.73 0.28 0.09 1.8 0.22 0.08 1.88 0.17 0.07 1.95 0.13 0.06 2.03 0.1 0.05 2.1 0.07 0.04 2.18 0.05 0.04 2.25 0.04 0.03 2.33 0.02 0.03 2.4 0.02 0.02 2.48 0.01 0.02 2.55 0.01 0.02 2.63 0 0.01 2.7 0 0.01 2.78 0 0.01 2.85 0 0.01 2.93 0 0.01 3 0 0
-4 -3 -2 -1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f(X) or f(Y) f(Z) Change the values in blue to see how the mean μ & the standard deviation σ control the curve. ( 29 2 2 2 2 1 ) ( σ μ π - - = x e x f

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Page 9 Normal Parameters: Mean = 0 Standard Deviation = 1 The probability of being between -10 and 1.96 is 0.98 Z = Z values are -10 and 1.96 X f(X) Area -3.5 0 0 0 -3.5 0 0 0 -3.49 0 0 -0.01 -3.49 0 0 0 -3.48 0 0 -0.01 -3.48 0 0 0 -3.47 0 0 -0.01 -3.47 0 0 0 -3.46 0 0 -0.01 -3.46 0 0 0 -3.45 0 0 -0.01 -3.45 0 0 0 -3.44 0 0 -0.01 -3.44 0 0 0 -3.43 0 0 -0.01 -3.43 0 0 0 -3.42 0 0 -0.01 -3.42 0 0 0 -3.41 0 0 -0.01 -3.41 0 0 0 -3.4 0 0 -0.01 -3.4 0 0 0 -3.39 0 0 -0.01 -3.39 0 0 0 -3.38 0 0 -0.01 -3.38 0 0 0 -3.37 0 0 -0.01 -3.37 0 0 0 -3.36 0 0 -0.01 -3.36 0 0 0 -3.35 0 0 -0.01 -3.35 0 0 0 -3.34 0 0 -0.01 -3.34 0 0 0 -3.33 0 0 -0.01 -3.33 0 0 0 -3.32 0 0 -0.01 -3.32 0 0 0 -3.31 0 0 -0.01 -3.31 0 0 0 -3.3 0 0 -0.01 -3.3 0 0 0 -3.29 0 0 -0.01 -3.29 0 0 0 -3.49 -3.31 -3.13 -2.95 -2.77 -2.59 -2.41 -2.23 -2.05 -1.87 -1.69 -1.51 -1.33 -1.15 -0.97 -0.79 -0.61 -0.43 -0.25 -0.07 0.11 0.29 0.47 0.65 0.83 1.01 1.19 1.37 1.55 1.73 1.91 2.09 2.27 2.45 2.63 2.81 2.99 3.17 3.35 0 0.05 0.1 0.15 0.2

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## This note was uploaded on 06/20/2011 for the course MGMT 524 taught by Professor Andrews,r during the Spring '08 term at VCU.

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Class 4 - Normal Distribution - For Continuous variables,...

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