# Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

In [1, p. 61 exercise 3.5] we are asked to show that for the conditions of Problem [1, p. 60, exercise 3.4] the CR bound is
 $var(\hat{\mu}_x) \geq \frac{\sigma_x^2}{N}$

and further to give a statement about the efficiency of the sample mean estimator. read the conclusion >
In [1, p. 60 exercise 3.4] we are asked to prove that the estimate
 $\hat{\mu}_x = \frac{1}{N}\sum\limits^{N-1}_{n=0}x[n]$ (1)

is an unbiased estimator, given $\left\{x[0],x[1],...,x[N]\right\}$ are independent and identically distributed according to a $N(\mu_x,\sigma^{2}_x)$ distribution. Furthermore we are asked to also find the variance of the estimator. read the conclusion >
In [1, p. 60 exercise 3.3] we are asked to prove that the complex multivariate Gaussian PDF reduces to the complex univariate Gaussian PDF if N=1. read the conclusion >
In [1, p. 60 exercise 3.2] we are asked to proof by using the method of characteristic functions that the sum of squares of N independent and identically distributed N(0,1) random variables has a $\chi^{2}_{N}$ distribution. read the conclusion >
In [1, p. 60 exercise 3.1] a $2 \times 1$ real random vector $\mathbf{x}$ is given , which is distributed according to a multivatiate Gaussian PDF with zero mean and covariance matrix:
 $\mathbf{C}_{xx}= \left[ \begin{array}{cc} \sigma_{1}^2 & \sigma_{12} \\ \sigma_{21} & \sigma_{2}^2 \end{array}\right]$

We are asked to find $\alpha$ if $\mathbf{y}=\mathbf{L}^{-1}\mathbf{x}$ where $\mathbf{L}^{-1}$ is given by the relation:

 $y_{1}$ $=$ $x_{1}$ $y_{2}$ $=$ $\alpha x_{1}+x_{2}$

so that $y_{1}$ and $y_{2}$ are uncorrelated and hence independent. We are also asked to find the Cholesky decomposition of $\mathbf{C}_{xx}$ which expresses $\mathbf{C}_{xx}$ as $\mathbf{L}\mathbf{D}\mathbf{L}^{T}$, where $\mathbf{L}$ is lower triangular with 1′s on the principal diagonal and $\mathbf{D}$ is a diagonal matrix with positive diagonal elements. read the conclusion >