# Lysario – by Panagiotis Chatzichrisafis

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

The desire to predict the complex WSS random process based on the sample $x[n-1]$ by using a linear predictor $\hat{x}[n]=-\alpha_{1}x[n-1]$ (1)

is expressed in [1, p. 61 exercise 3.10]. It is asked to chose $\alpha_{1}$ to minimize the MSE or prediction error power $MSE = \mathcal{E}\left\{\left| x[n] -\hat{x}[n] \right|^{2} \right\}.$ (2)

We are asked to find the optimal prediction parameter $\alpha_{1}$ and the minimum prediction error power by using the orthogonality principle.
In [1, p. 61 exercise 3.9] we are asked to consider the real linear model $x[n]=\alpha + \beta n + z[n] \; n=0,1,...,N-1$ (1)

and find the MLE of the slope $\beta$ and the intercept $\alpha$ by assuming that $z[n]$ is real white Gaussian noise with mean zero and variance $\sigma_{z}^{2}$. Furthermore it is requested to find the MLE of $\alpha$ if in the linear model we set $\beta=0$. read the conclusion >
In [1, p. 61 exercise 3.8] we are asked to prove that the sample mean is a sufficient statistic for the mean under the conditions of [1, p. 61 exercise 3.4]. Assuming that $\sigma^{2}_{x}$ is known. We are asked to find the MLE of the mean by maximizing $p(\hat{\mu}_{x},\mu_{x})$. read the conclusion >
In [1, p. 61 exercise 3.7] we are asked to find the MLE of $\mu_{x}$ and $\sigma_x^2$. for the conditions of Problem [1, p. 60 exercise 3.4] (see also [2, solution of exercise 3.4]). We are asked if the MLE of the parameters are asymptotically unbiased , efficient and Gaussianly distributed.
In [1, p. 61 exercise 3.6] we are asked to assume that the variance is to be estimated as well as the mean for the conditions of [1, p. 60 exercise 3.4] (see also [2, solution of exercise 3.4]) . We are asked to prove for the vector parameter $\mathbf{\theta}=\left[\mu_x \; \sigma^2_x\right]^T$, that the Fisher information matrix is $\mathbf{I}_{\theta}=\left[\begin{array}{cc} \frac{N}{\sigma^2_x} & 0 \\ 0 & \frac{N}{2\sigma^4_x} \end{array}\right]$
Furthermore we are asked to find the CR bound and to determine if the sample mean $\hat{\mu}_x$ is efficient. If additionaly the variance is to be estimated as $\hat{\sigma}^2_x=\frac{1}{N-1}\sum\limits_{n=0}^{N-1}(x[n]-\hat{\mu}_x)^2$ $\frac{(N-1)\hat{\sigma}^2_x}{\sigma^2_x} \sim \chi^2_{N-1}$