MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Random Vectors and Random Sampling. 1+ x2 +y 2 ) (n+2)/2

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1 MATH 3806/MATH4806/MATH6806: MULTIVARIATE STATISTICS Solutions to Problems on Rom Vectors Rom Sampling Let X Y have the joint pdf: fx,y) + x +y ) n+)/ π n for < x < < y < this is particular case of the multivariate t distribution - see Kotz Nadarajah 004) Multivariate t Distributions Their Applications Cambridge University Press) Since fx,y) is an even function with respect to x y, we have EX) ) x + x +y n+)/ dydx 0, π n EY) y + x +y ) n+)/ dydx 0, π n EXY) ) xy + x +y n+)/ dydx 0, π n so CovX,Y) EXY) EX)EY) 0 The marginal pdf of X is f X x) + x +y ) n+)/ dy π n Similarly, the marginal pdf of Y is n n/+ πn+x ) n+)/ v n )/ v) / dv 0 n n/+ n+ πn+x ) n+)/b, ) Γn+)/) nπγn/) +x /n) n+)/ f Y y) Γn+)/) nπγn/) +y /n) n+)/ Note that fx,y) f X x)f Y y), so X Y are not independent Let X Y be rom variables such that the pdf of Y is: fy) y exp y )

2 the conditional pdf of X given Y is: fx y) ) yexp yx π Since fx y) is an even function with respect to x, we have EX Y) xfx y) ) y xexp yx 0 π So, EEX Y) E0 0 However, the marginal pdf of X is f X x) fy)fx y) π exp y x + ) ) dy 0 which is the Cauchy pdf So, EX) does not exist 3 Let 0 X T πx +), be a 3 data matrix of observations on the rom vector X T X,X ) Consider the linear combinations Y X +X Y X +3X i) The sample means for X X are 6/3 5/3 5 The sample variance for X X are /){ +3 + ) 3 } /){ ) 3 5 } 6 The sample covariance between X X is /){9+5+) 3 5} So, x 5 S X 6 ii) Note that Y T Thesamplemeans for Y Y are 4/3 8 57/3 9 Thesample variance for Y Y are/){ ) 3 8 } 73 /){ ) 3 9 } 4 Thesample covariance between Y Y is /){ ) 3 8 9} 9 So, 8 ȳ 9 S Y

3 iii) Note that Y Y 3 X X So, the mean vector covariance matrix of Y are ȳ x S y Suppose that X,X X 3 are independent rom variables with unit variance Let Y X +X +X 3, Y X X Y 3 X X 3 Note that Y Y Y X X X 3 So, the covariance correlation matrices of Y T Y,Y,Y 3 ) are , respectively 5 Given a data matrix, X the resulting sample correlation matrix, R, consider the stardized observations x ij x j )/ s jj, i,,n; j,,p Let z ij x ij x j )/ s jj Then z ij i 0, n ) ) x ij n x j sjj i sjj n x j n x j ) 3

4 n z ij z j ) i n n s jj n s jj s jj zij i x ij x j ) /s jj i x ij x j ) i n z ij z j )z ik z k ) i n n z ij z ik i x ij x j )x ik x k )/{ s jj s kk } i x ij x j )x ik x k ) n i sjj s kk s jk sjj s kk r jk Hence, the stardized quantities have sample covariance matrix, R 6 The following data correspond to used-car prices for a sample of cars of a particular type Age, X is measured in years selling price, X is measured in thouss of pounds i) The scatterplot of the data: x x

5 Prize Age There is an inverse linear relationship between age prize ii) For the data, x 7,098) T, S R The negative correlation coefficient quantifies the strength of the inverse linear trend iii) TheeigenvaluesofSare6398, 0509 Thecorrespondingeigenvectors are 09764,06 T, 06, T So, S / S / / / iv) TheEuclideistancebetweenx T 3,3) xt 0,03) is 3) +03 3) 85 The Mahalanobis distance between x T 3,3) xt 0,03) is , ) 8, ) T , ) 8, ) T

6 7 The sample covariance matrix corresponding to 9 5 X 3 is So, S 6 4 S 6 8 Let V be a rom vector with EV) µ EV µ)v µ) T ) Σ Then E VV T) E V µ+µ)v µ+µ) T) E V µ+µ) V µ) T +µ T)) E V µ)v µ) T +V µ)µ T +µv µ) T +µµ T) E V µ)v µ) T) +E V µ)µ T) +E µv µ) T) +µµ T Σ+µµ T 9 EX+Y) EX)+EY) since the i,j)th element of EX+Y) is EX ij +Y ij ) EX ij )+EY ij ), which is the i,j)th element of EX)+EY) EAXB) AEX)B since the i,j)th element of EAXB) is E A il X lk )B kj Ail EX lk )B kj, which is the i,j)th element of AEX)B 0 Suppose the rom variables X Y have the joint pdf { fx,y) π) 3/ x +y exp x +y } Since this function is even with respect to x y, we have { EXY) xy π) 3/ x +y exp x +y } dxdy 0, EX) { x π) 3/ x +y exp x +y } dxdy 0, EY) { y π) 3/ x +y exp x +y } dxdy 0, so CovX,Y) EXY) EX)EY ) 0 However, the given form for fx,y) does not factor out into components for x y So, we have an example where CovX,Y) 0 does not imply independence 6

7 Let X have the covariance matrix Σ Then ρ V /5 4/5 /5 /6 4/5 /6 / / /9 Also Corr X,X +X 3 )/) /) Corr X,X )+/) Corr X,X 3 ) /0+/5 /30 For X X, we have EX X ) EX ) EX ) VarX X ) VarX )+ 4VarX ) 4CovX,X ) For X + 3X, we have E X + 3X ) EX ) + 3EX ) Var X + 3X ) VarX )+9VarX ) 6CovX,X ) For X +X +X 3, we have EX +X +X 3 ) EX )+EX )+EX 3 ) VarX +X + X 3 ) VarX )+VarX )+VarX )+CovX,X )+CovX,X 3 )+CovX,X 3 ) For X +X X 3, we have EX +X X 3 ) EX )+EX ) EX 3 ) VarX + X X 3 ) VarX )+4VarX )+VarX 3 )+4CovX,X ) CovX,X 3 ) 4CovX,X 3 ) For 3X 4X, wherex X are independent, we have E3X 4X ) 3EX ) 4EX ) Var3X 4X ) 9VarX )+6VarX ) 3 Lettheromvector X T X,X,X 3,X 4 )withmeanvector µ T 4,3,,) variancecovariance matrix Σ Partition X as X X X X 3 X 4 X ) X ) Let A, 7

8 B EX ) ) 4,3 T EAX ) ),4,3 T 0 Cov X )) Cov AX )) 3 0, 0, T 7 EX ) ), T E BX )), T,3 T Cov X )) 9 4 Cov BX )) Cov X ),X )) 0 Cov AX ),BX )), 0 4, 0,6 4 Let e i denote the unit vector with in the ith place 0 elsewhere Let Y j, j,, be a sequence of independent identically distributed rom vectors such that PrY e i ) a i, i,,p, where a i > 0 a i Then EY) a e +a e + +a p e p a 8

9 Further, if X n j Y j then CovY) E Y a)y a) T E Y a)y T a T ) E YY T Ya T ay T +aa T E YY T aa T a e e T +a e e T + +a pe p e T p aat diaga) aa T EX) E Y j EY j ) na j j VarX) Var Y j VarY j ) n diaga) aa T) j j Finally, note that diaga) aa T ) diaga) aa T a a a i a a 0 So, VarX) is singular 9