W is a linear combination of normal random variables

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W is a linear combination of normal random variables and therefore W also follows a normal distribution. The mean of W is found as: 1700 40 30 25 20 30 20 = + = + = ) Y ( E ) X ( E ) W ( E
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Econ 325 – Chapter 5 35 Recall the definition of correlation: Y X XY ) Y , X ( Cov σ σ = ρ By rearranging the covariance can be calculated as: 39.6 121 81 0.4 - = - = σ σ ρ = Y X XY ) Y , X ( Cov The variance of W is found as: 93780 39.6 30 20 2 121 30 81 20 30 20 2 30 20 2 2 2 2 = - + = + + = ) Y , X ( Cov ) Y ( Var ) X ( Var ) W ( Var The standard deviation of W is: 306.235 93780 = = σ W Econ 325 – Chapter 5 36 To find ) W ( P 2000 < with Microsoft Excel select Insert Function: NORMDIST( 2000, 1700, 306.235, 1 ) W μ W σ This returns the probability 0.8364 . Therefore, the probability that the portfolio value exceeds 2,000 is: 1 - 0.8364 = 0.16 Now check this result using the table for the standard normal distribution. Write the probability for W as a probability for Z : 0.8365 Table Appendix the in up look 0.98 0.98) 306.235 1700 2000 2000 2000 = = < = - < = σ μ - < σ μ - = < ) ( F Z ( P Z P W P ) W ( P W W W W The use of the standard normal distribution table may give slight rounding differences in results compared to Microsoft Excel.
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