Problem set 4: Gaussian Vectors
Let \(X=(\xi,\eta)\) be a Gaussian vector with covariance matrix \(\Sigma\).
Find \(a,b\) such that
\[ \eta+a\xi \qquad\text{and}\qquad \eta+b\xi \]
are independent random variables.
Also, consider the case when
\[ \Sigma=\sigma^2 I_n, \]
where \(I_n\) is the \(n\times n\) identity matrix, and \(C\) is an orthogonal matrix.
Let \(\xi,\eta\) be two random variables with normal distributions.
Is it always true that \((\xi,\eta)\) is a Gaussian vector?
Let \((X,Y)\) be a Gaussian vector, where
\[ X=2\xi+3\eta, \qquad Y=\xi-\eta. \]
Find the value of \(\operatorname{Cov}(\xi,\eta)\) for which the random variables \(X\) and \(Y\) are independent.
Assume that
\[ \operatorname{Var}(\xi)=\operatorname{Var}(\eta)=1. \]
Let \((X,Y)\) be a Gaussian vector with
\[ (X,Y)\sim\mathcal{N}(0,\Sigma), \]
where
\[ \Sigma= \begin{pmatrix} 3 & -1\\ -1 & 2 \end{pmatrix}. \]
Find
\[ \mathsf{E}[X^3Y]. \]
Let \((X_1,\ldots,X_n)\) be a sequence of i.i.d. random variables with
\[ X_i\sim\mathcal{N}(a,\sigma^2). \]
Let
\[ \overline{X}=\frac{1}{n}\sum_{i=1}^n X_i. \]
Prove that the random variable \(\overline{X}\) and the random vector
\[ (X_1-\overline{X},\ldots,X_n-\overline{X}) \]
are independent.
Let \(\xi_1,\xi_2,\ldots,\xi_n\) be independent random variables with
\[ \xi_i\sim\mathcal{N}(0,\sigma_i^2), \qquad i=1,\ldots,n. \]
Prove that the random vector
\[ \eta= (\xi_1,\, \xi_1+\xi_2,\, \ldots,\, \xi_1+\xi_2+\cdots+\xi_n) \]
is Gaussian and find its distribution.
Let \((X,Y,Z)\) be a Gaussian vector.
Find
\[ \mathsf{E}[X^2YZ]. \]
Let \(\xi_1,\xi_2,\ldots\) be a sequence of independent identically distributed random variables with distribution
\[ U[0,1]. \]
Set
\[ X_i=(\xi_i,\xi_i^2). \]
Find the limit in distribution of the random vectors
\[ \frac{ X_1+\cdots+X_n- \left(\frac n2,\frac n3\right) }{\sqrt n}. \]
Let \((X,Y)\) be a Gaussian vector with
\[ (X,Y)\sim\mathcal{N}(0,\Sigma), \]
where
\[ \Sigma= \begin{pmatrix} 3 & -1\\ -1 & 2 \end{pmatrix}. \]
Find
\[ \mathsf{E}[Ye^X]. \]
Let \((\xi,\eta)\) be a Gaussian vector such that
\[ \mathsf{E}\xi=\mathsf{E}\eta=0, \]
\[ \operatorname{Var}(\xi)=\operatorname{Var}(\eta)=2, \qquad \mathsf{E}[\xi\eta]=1. \]
Find the density of
\[ \arctan\left(\frac{\xi}{\eta}\right). \]