Question: If \(X_{i}, i=1,2, \ldots, n\), are independent random variables each with zero mean, and \(Z=\sum_{i=1}^{n} X_{i}\), find:

1. \(E\{Z\}\).
2. \(E\left\{Z^{2}\right\}\) and \(\sigma_{z}^{2}\).
3. If \(X_{i}\) is \(\mathrm{U}(-1 / 2,1 / 2)\), what are the values of \(E\{Z\}, E\left\{Z^{2}\right\}\) and \(\sigma_{z}^{2}\) ?
4. If \(X_{i}\) is \(\mathrm{U}(0,1)\) and \(n \geq 12\), write a very simple mathematical formula that is a function of \(Z\) and may be used to express an approximate Gaussian RV with arbitrary mean and variance. (Hint: This is very similar to the u-substitution used to transform a normalized Gaussian to an arbitrary Gaussian.)

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