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

(a) Find the moment-generating function of X.

(b) Using this moment generating function verify that the mean of the random variable is \(\mu \) and the variance is \({{\sigma }^{2}}\).

(c) Suppose that \(Y=\frac{{{\left( X-\mu \right)}^{2}}}{{{\sigma }^{2}}}\). Verify that Y has a \({{\chi }^{2}}\) distribution with 1 degree of freedom.