STAT 501 Assignment 1 Name Spring 2005

Transcription

1 STAT 50 Assignment Name Spring 005 Reading Assignment: Johnson and Wichern, Chapter, Sections.5 and.6, Chapter, and Chapter. Review matrix operations in Chapter and Supplement A. Written Assignment: Due Monday, January, in class.. Consider a random vector X = (X X X ) with mean vector µ = (,, ) and covariance matrix Σ. The eigenvalues of Σ are λ =, λ = 6, λ = and the corresponding eigenvectors are / / 6 0 e = / e = / 6 e = / / / 6 / Evaluate the following uantities. Parts (a), (b), (c), (d) and (g) should be done without using a computer. (a) Σ = (b) trace(σ) = (c) V(e X) = (d) = (e) = (f) / =

2 (g) Define Y = e X, Y = e X, and Y = e X. Find the mean vector and covariance matrix for Y = (Y Y Y ) =Γ X, where the i-th column of Γ is the i-th eigenvector for Σ. Evaluate E( Y) = V(Y) =Γ Γ= (h) The linear transformation, Y = Γ X, examined in part (g) is often called a rotation because it corresponds to simply rotating the coordinate axes. Use the definition of eigenvectors to show that lengths of vectors are not changed by this transformation, i.e., show Y Y = X X when Y = Γ X. (i) Let Γ be a matrix for which the i-th column of Γ is the i-th eigenvector for Σ, and let Y =Γ X,, µ X = E(X) X Σ = V(X), µ Y =Γµ X, and Y = V(Y) = Γ XΓ. Determine which of the following measures of variability or distance are unaffected by rotations. Justify your answers by either giving a counter example or a proof using the properties of eigenvalues, eigenvectors, and matrix operations given in chapter of Johnson and Wichern, (a) Is Σ X = Σ Y? (b) Is trace (Σ X ) = trace (Σ Y )? (c) Is Σ X = Σ Y? (d) Is (X X) X µ (X µ X) = (Y µ Y) Y (Y µ Y)? (e) Is (X µ X) (X µ X) = (Y µ Y) (Y µ Y)?. Consider a bivariate normal population with mean vector µ and covariance matrix Σ, where 5 µ = and = 9 (a) Write down a formula for the joint density function.

3 (b) Find the eigenvalues and eigenvectors for Σ. λ = λ = e = e = (c) Find values for Σ = trace (Σ) = (d) Write down the formula for the boundary of the smallest region such that there is probability 0.5 that a randomly selected observation will be inside the boundary. (e) Sketch the boundary of the region you identified in part (d). X X - (f) (g) Determine the area of the region described in parts (d) and (e). Determine the area of the smallest region such that there is probability 0.95 that a randomly selected observation will be in that region.. The joint density function for and X is X f (x, x ) = π exp{.5x + x x.5x x + 7x.5} Find the mean vector and the covariance matrix for this bivariate distribution.

4 . Let X be a normally distributed random vector with 0 µ = and = (a) Which of the following are pairs of independent random variables? (i) X and X (iv) (X X ) and, X (ii) X and X (v) X + X and X - X (iii) X and X + X X (vi) X + X + X and X - X + X (b) What is the distribution of Y = (X,X )? (c) What is the conditional distribution of Y = (X,X ) (d) given X = x? Find the correlation between X and X and a formula for the partial correlation between X and X given X = x. ρ = ρ = (e) What is the conditional distribution of X given X = x and X = x? 5. Suppose X N µ, where µ = = (a) What is the distribution of Z = X 5X + X? (b) What is the joint distribution of Z in part (a) and Z = X X + X. (c) X Find the conditional distribution of X given X =. (d) Find the partial correlation between X and X given X = x and X = x.

5 5 6. Let and X be N µ be where and, and Y N µ, X Y are independent µ = 6 8 = µ = = (a) Evaluate Cov X Y, X Y + (b) Are X Y and X + Y independent random vectors? Explain. (c) Show that the joint distribution for the four dimensional random vector X X + Y Y is a multivariate normal distribution. 7. Let d (p,) denote a measure of distance between p and. Johnson and Wichern indicate that any distance measure should possess the following four properties: (i) d (p, ) = d (,p) (ii) d (p, ) > 0 if p (iii) d(p, ) = 0 if p = (iv) d (p, ) (p, r) + d(r,), where r = ( r,r )'. d These properties are satisfied, for example, by Euclidean distance, i.e.,. d(p, ) = [ (p )' (p )] / (a) In this class we will often consider distance measures of the form d(p, ) = [ (p )' A(p )] /. Sometimes A will be the inverse of a covariance matrix which implies that A is symmetric and positive definite. Show that properties (i) through (iv) are satisfied when A is symmetric and positive definite.

6 6 (b) Does A have to be both symmetric and positive definite for d(p, ) = [ (p )' A(p )] / to satisfy properties (i) through (iv)? Present your proofs, counter examples or explanations. (c) For k-dimensional vectors it is obvious that the measure d (p,) = max { p, p,..., p } satisfies properties (i), (ii), (iii). Either prove or disprove that it satisfies property (iv), the triangle ineuality. k k 8. (a) By multiplication of partitioned matrices, verify for yourself that I B I B = 0 I 0 I r r r r r r and r r r r r r I 0 I 0 = B I B I for any r matrix B, where 0 a b denotes a matrix of zeros and I a adenotes an identity matrix. (b) Show that I B Ir r 0r = = for any r matrix B. 0 I B I r r r (c) By multiplication of partitioned matrices verify that if A A A = A A is a suare matrix for which A is matrix and A is an r r matrix and the inverse of A exists, then I r 0 r 0 A r A 0r Ir r A A A Ir r A A A A 0 I A A A A = (d) Use the results from parts (a) through (c) to show that A A A A AA = for any suare matrix A with A 0.

7 7 (e) Use the results from parts (a) through (d) to show that ( ) I 0 r A A A A 0 r I AA A = A A Ir r 0 0r Ir r r A for any symmetric matrix A with A Consider a multivariate normal distribution with positive definite covariance matrix =. Note that from part (d) in problem 8, =, the product of determinants of covariance matrices for a marginal and a conditional normal distribution. Use the result from part (e) in problem 8 to expand x x µ µ (x µ ) (x µ ) = x µ x µ in an appropriate way to show that the multivariate normal density function is a product of the density functions for the marginal distribution of X and a conditional distribution. What happens when = 0, i.e. when X and X are uncorrelated? For additional practice you could do problems.6,.8,.9,.0,.,.6 at the end of Chapter and problems.,.,.,.5,.,.5,.6,.7 at the end of Chapter. Problem.8 gives a trivial example of a non-normal bivariate distribution with normal marginal distributions. We will consider a more substantial example on the next assignment. Do not hand in these additional problems; answers will be given on the solution sheet for this assignment.