Chapter 9: Problem 13
Fit by the method of least squares the plane \(z=a+b x+c y\) to the five points \((x, y, z):(-1,-2,5),(0,-2,4),(0,0,4),(1,0,2),(2,1,0)\).
Chapter 9: Problem 13
Fit by the method of least squares the plane \(z=a+b x+c y\) to the five points \((x, y, z):(-1,-2,5),(0,-2,4),(0,0,4),(1,0,2),(2,1,0)\).
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Get started for freeTwo experiments gave the following results: $$ \begin{array}{cccccc} \hline \mathrm{n} & \bar{x} & \bar{y} & s_{x} & s_{y} & \mathrm{r} \\ \hline 100 & 10 & 20 & 5 & 8 & 0.70 \\ 200 & 12 & 22 & 6 & 10 & 0.80 \\ \hline \end{array} $$ Calculate \(r\) for the combined sample.
Let the independent random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\beta x_{i}, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one is zero. Find the maximum likelihood estimators of \(\beta\) and \(\gamma^{2}\).
A random sample of size \(n=6\) from a bivariate normal distribution yields a value of the correlation coefficient of \(0.89 .\) Would we accept or reject, at the 5 percent significance level, the hypothesis that \(\rho=0\).
Let \(\mathbf{A}=\left[a_{i j}\right]\) be a real symmetric matrix. Prove that \(\sum_{i} \sum_{j} a_{i j}^{2}\) is equal to the sum of the squares of the eigenvalues of \(\mathbf{A}\). Hint: If \(\boldsymbol{\Gamma}\) is an orthogonal matrix, show that \(\sum_{j} \sum_{i} a_{i j}^{2}=\operatorname{tr}\left(\mathbf{A}^{2}\right)=\operatorname{tr}\left(\mathbf{\Gamma}^{\prime} \mathbf{A}^{2} \mathbf{\Gamma}\right)=\) \(\operatorname{tr}\left[\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \mathbf{\Gamma}\right)\left(\mathbf{\Gamma}^{\prime} \mathbf{A} \boldsymbol{\Gamma}\right)\right]\)
Suppose \(\mathbf{A}\) is a real symmetric matrix. If the eigenvalues of \(\mathbf{A}\) are only 0 's and 1 's then prove that \(\mathbf{A}\) is idempotent.
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