Question: Suppose that the random variable \(X\tilde{\ }N\left( \mu ,{{\sigma }^{2}} \right)\)

1. Find the moment-generating function of X .
2. Using this moment generating function verify that the mean of the random variable is \(\mu \) and the variance is \({{\sigma }^{2}}\).
3. Suppose that \(Y=\frac{{{\left( X-\mu \right)}^{2}}}{{{\sigma }^{2}}}\). Verify that Y has a \({{\chi }^{2}}\) distribution with 1 degree of freedom.

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