Math Problems Vault

Problem 2a.1

(a)(b)

Show that

f\left(y_{1}, y_{2}\right) = k^{-1} \exp \left[-\frac{1}{2}\left(2 y_{1}^{2}+y_{2}^{2}+2 y_{1} y_{2}-22 y_{1}-14 y_{2}+65\right)\right]

is the density of a bivariate normal random vector $\mathbf{Y} = \left(Y_{1}, Y_{2}\right)^{T}$ .

?

(a)

Find $k$ .

(b)

Find $\mathbb {E}\left[\mathbf{Y}\right]$ and $\operatorname {Var}\left[\mathbf{Y}\right]$ .

Problem 2a.2

Let $\mathbf{U}$ have density $g$ and let $\mathbf{Y} = \mathbf{A}(\mathbf{U}+\mathbf{c})$ , where $\mathbf{A}$ is nonsingular. Show that the density $f$ of $\mathbf{Y}$ satisfies

f(\mathbf{y}) = g(\mathbf{u}) /\left|\operatorname {det}\left(\mathbf{A}\right)\right|

where $\mathbf{y} = \mathbf{A}(\mathbf{u}+\mathbf{c})$ .

?

Problem 2a.3

(a)(b)

?

(a)

Show that the $3 \times 3$ matrix

\mathbf{\Sigma } =\left[\begin{smallmatrix} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{smallmatrix}\right]

is positive-definite for $\rho > -\frac{1}{2}$ .

(b)

Find $\mathbf{\Sigma }^{1 / 2}$ when

\mathbf{\Sigma } =\left[\begin{smallmatrix} 1 & \rho \\ \rho & 1 \end{smallmatrix}\right]