Suppose X = (X1, X2, X3, X4)T has a multivariate normal N4(0, I2 ⊗ Σ), where I2 is the 2 × 2 identity matrix, ⊗ is the Kronecker product, and Σ = [ 2 -1][-1 2]. Define Z = [X1 X2][X3 X4] and Q = ((Qij)) = ZTZ. Suppose χn2 denotes a chi-square random variate with n degrees of freedom, and Wm(n, Σ) denotes a Wishart distribution of order m with parameters n and Σ. The distribution of (Q11 + Q12 + Q21 + Q22) is:

Let X = (X1, X2, X3)T be a 3 × 1 random vector with E(X) = (3, 2, 1)T and covariance matrix Cov(X) = Σ = 3-20-23-20-23. Suppose that Y = (Y1, Y2, Y3)T = 1√3X. Then the value of the multiple correlation coefficient between Y1 and (Y2, Y3) equals:

Suppose the column vector [X1, X2, X3]T follows a multivariate normal distribution N3(0, I), where 0 is the zero vector and I is the identity matrix. If [X, Y, Z]T is defined by the matrix transformation with rows [3, 0, 0], [2, 2, 0], and [4, 0, 4] multiplied by [X1, X2, X3]T, then the partial correlation coefficient ρyz.x is: