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**Unformatted text preview: **2. If someone were to suggest that you estimate the mean ˆ μ and the variance ˆ σ 2 of the log return of your portfolio so that you could be sure that your log return belongs to the interval [ˆ μ- 3ˆ σ, ˆ μ + 3ˆ σ ] with probability at least 99% , what kind of assumption would this someone working under, and which of the three computational algorithms of question 1 would this someone be closest to? S Problem 3.4 1. Construct a vector of 100 increasing and regularly spaced numbers starting from . 1 and ending at 20 . Call it SIG2 . Construct a vector of 21 increasing and regularly spaced numbers starting from- 1 . and ending at 1 . . Call it RHO . 2. For each entry σ 2 of SIG2 and for each entry ρ of RHO : • Generate a sample of size N = 500 from the distribution of a bivariate normal vector Z = ( X,Y ) , where X ∼ N (0 , 1) , and Y ∼ N (0 ,σ 2 ) , and the correlation coefficient of X and Y is ρ (the S object you create to hold the values of the sample of Z ’s should be a 500 × 2 matrix); 166 3 MULTIVARIATE DATA EXPLORATION • Create a 500 × 2 matrix, call it EXPZ , with the exponentials of the entries of Z (the distributions of these columns are lognormal as defined in Problem 3.12);‘ • Compute the correlation coefficient, call it ˜ ρ , of the two columns of EXPZ 3. Produce a scatterplot of all the points ( σ 2 , ˜ ρ ) so obtained. Comment. T Problem 3.5 Let X and Y be continuous random variables with c.d.f.s F X and F Y respec- tively, and with copula C . For each real number t , prove the following two equalities: 1 . P { max( X,Y ) ≤ t } = C ( F X ( t ) ,F Y ( t )) 2 . P { min( X,Y ) ≤ t } = F X ( t ) + F Y ( t )- C ( F X ( t ) ,F Y ( t )) T Problem 3.6 Suppose X 1 and X 2 are independent, and X 1 ∼ N (0 , 1) , X 2 ∼ N (0 , 1) . Define X 3 = | X 2 | , if X 1 >-| X 2 | , if X 1 < 1. Compute the cdf of X 3 . Conclude if X 3 is normal. 2. Compute P { X 2 + X 3 = 0 } . 3. Is X 2 + X 3 normal? Is ( X 2 ,X 3 ) jointly normal? T Problem 3.7 Let us assume that X 1 , X 2 and X 3 are independent N (0 , 1) random variables and let us set Y 1 = X 1 + X 2 + X 3 √ 3 and Y 2 = X 1- X 2 √ 2 1. Compute cov ( Y 1 ,Y 2 ) 2. Compute var ( Y 1 Y 2 ) T Problem 3.8 This elementary exercise is intended to give an example showing that lack of correlation does not necessarily mean independence! Let us assume that X ∼ N (0 , 1) and let us define the random variable Y by: Y = 1 p 1- 2 /π ( | X | - p 2 /π ) 1. Compute E {| X |} 2. Show that Y has mean zero, variance 1 , and that it is uncorrelated with X . T Problem 3.9 Let us assume that X 1 and X 2 are independent N (0 , 1) random variables and let us define the random variable Y by Y = | X 2 | , if X 1 >-| X 2 | , if X 1 ≤ 1. Prove that Y ∼ N (0 , 1) 3. Say if ( X 1 ,Y ) is bivariate Gaussian, and explain why. Problems 167 T Problem 3.10 The purpose of this problem is to show that lack of correlation does not imply independence, even when the two random variables are Gaussian !!!...

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