Test after the course by A.Zaigraev

”Multivariate Distributions in Statistics”,

(20.05.2003, Tampere)

1. Let the density function of a bivariate normal distribution be of the form

f(x_{1}, x_{2}) = \frac{1}{6π} exp \{−\frac{1}{18} (5x_{1}^{2} + 2x_{1}x_{2} + 2x_{2}^{2} − 8x_{1} + 2x_{2} + 5)\} .

Find μ, Σ and the correlation coefficient ρ.

2. Calculate P (X_{1} > 0, X_{2} < 0) for a standard bivariate normal vector X = (X_{1}, X_{2})′ with the correlation coefficient ρ.

3. Assume that X = (X_{1}, X_{2}, X_{3})′ ∼ N_{3}(μ, Σ) with

μ = ( \begin{matrix}3 \\ 4 \\ 1 \end{matrix} ) , Σ = ( \begin{matrix}1 & 3 & 1 \\ 3 & 10 & 2 \\ 1 & 2 & 4 \end{matrix} ) .

Find the conditional distribution of X_{1} given (X_{2}, X_{3})′ = (x_{2}, x_{3})′, the multiple correlation coefficient R_{1·23} and the partial correlation coefficient ρ_{12·3}.

4. Write down the density function of the Dirichlet distribution in the case d = 2, n_{0} = n_{1} = n_{2} = 1. What distribution do we obtain (it is known for you)?