Question

Let $\mathbf{X}=(X, Y)$ be a jointly Gaussian random vector with zero means, unit variances, and unknown correlation coefficient $\rho$. Consider a random sample of $n$ such vectors. (a) Show that the ML estimator for $\rho$ involves solving a cubic eqation. (b) Show that Problem 8.23 gives the ML estimator if the mean and variances are unknown. (c) Repeat 5 trials of the following: Generate a sample of 100 observations of the pairs of zero-mean, unit-variance Gaussian random variables and estimate $\rho$. using parts a and b for the cases: $\rho=0.5, \rho=0.9$, and $\rho=0$.

   Let $\mathbf{X}=(X, Y)$ be a jointly Gaussian random vector with zero means, unit variances, and unknown correlation coefficient $\rho$. Consider a random sample of $n$ such vectors.
(a) Show that the ML estimator for $\rho$ involves solving a cubic eqation.
(b) Show that Problem 8.23 gives the ML estimator if the mean and variances are unknown.
(c) Repeat 5 trials of the following: Generate a sample of 100 observations of the pairs of zero-mean, unit-variance Gaussian random variables and estimate $\rho$. using parts a and b for the cases: $\rho=0.5, \rho=0.9$, and $\rho=0$.

Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia 3rd Edition

Chapter 8, Problem 33

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Step 1

Since $\mathbf{X}$ is a jointly Gaussian random vector with zero means and unit variances, we can express its covariance matrix as: \[ \Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \] where $\rho$ is the unknown correlation coefficient. Show more…

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Let $\mathbf{X}=(X, Y)$ be a jointly Gaussian random vector with zero means, unit variances, and unknown correlation coefficient $\rho$. Consider a random sample of $n$ such vectors. (a) Show that the ML estimator for $\rho$ involves solving a cubic eqation. (b) Show that Problem 8.23 gives the ML estimator if the mean and variances are unknown. (c) Repeat 5 trials of the following: Generate a sample of 100 observations of the pairs of zero-mean, unit-variance Gaussian random variables and estimate $\rho$. using parts a and b for the cases: $\rho=0.5, \rho=0.9$, and $\rho=0$.

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