Using the generating function \(Z\) introduced in an earlier problem, show that:

(a) If \(X\) is a stochastic variable with a gaussian distribution with mean \(x_{0}\) and variance \(\sigma^{2}\), then \(a+b X\) is a stochastic variable with a Gaussian distribution with mean \(a+b x_{0}\) and variance \(b^{2} \sigma^{2}\).

(b) If \(X\) and \(Y\) are Gaussian stochastic variables with means \(x_{0}\) and \(y_{0}\) respectively, and variances \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\), then \(X+Y\) is a Gaussian stochastic variable with mean \(x_{0}+y_{0}\) and variance \(\sigma_{X}^{2}+\sigma_{Y}^{2}\).