Isye 2027

# 35 by lotus 1 2v 1 e y 2 2v 2 dudv v 1 0 1 0 1 2v

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Unformatted text preview: D (1) = 0.4, pD (0.99) = 0.1. The CLT pertains to sums of random variables, so we will apply the Gaussian approximation to ln(Y ), which is a sum of a large number of random variables: 365 ln Y = ln Dk . k=1 By LOTUS, the mean and variance of ln(Dk ) are given by µ = E [ln(Dk )] = 0.5 ln(1.01) + 0.4 ln(1) + 0.1 ln(0.99) = 0.00397. 2 σ 2 = Var(Dk ) = E [Dk ] − µ2 = 0.5(ln(1.01))2 + 0.4(ln(1))2 + 0.1(ln(0.99))2 − µ2 = 0.00004384. √ Thus, ln Y is approximately Gaussian with mean 365µ = 1.450 and standard deviation 365σ 2 = 0.127. Therefore, P {Y ≥ c} = P {ln(Y ) ≥ ln(c)} ln(c) − 1.450 ln(Y ) − 1.450 ≥ =P 0.127 0.127 ln(c) − 1.450 ≈Q . 0.127 In particular: (a) P {Y ≥ 3} ≈ Q(−2.77) ≈ 0.997. (b) P {Y ≥ 4} ≈ Q(−0.4965) = 0.69. (c) The median is the value c such that P {Y ≥ c} = 0.5, which by the Gaussian approximation is e1.450 = 4.26. (This is the same result one would get by the following argument, based on the law of large numbers. We expect, during the year, the stock to increase by one...
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