Chapter3

2 if someone were to suggest that you estimate the

This preview shows pages 47–50. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 !!!...
View Full Document

This note was uploaded on 09/11/2013 for the course ORF 405 taught by Professor Renecarmona during the Fall '10 term at Princeton.

Page47 / 53

2 If someone were to suggest that you estimate the mean ˆ...

This preview shows document pages 47 - 50. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online