Question: Let (Z) be a (mathrm{N}(0,1)) random variable. a. Prove that the moment generating function of (X) is (exp left[-frac{1}{2} t^{2}ight]). b. Prove that the characteristic

Let \(Z\) be a \(\mathrm{N}(0,1)\) random variable.

a. Prove that the moment generating function of \(X\) is \(\exp \left[-\frac{1}{2} t^{2}ight]\).

b. Prove that the characteristic function of \(X\) is also \(\exp \left[-\frac{1}{2} t^{2}ight]\).

c. Using the moment generating function, derive the first three moments of \(X\). Repeat the process using the characteristic function.

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