Consider a Gaussian distribution random variable X with mean $\mu_x$ and variance $\sigma_x^2$.
The MGF of X is defined as $M_x(t) = E[e^{tX}]$.
(a) Derive the MGF of X.
(b) If the first moment and second moment of X are 2 and 20, respectively, find the MGF
$M_z(t)$ of the random variable $Z = \frac{X+6}{2}$ based on the result obtain from (a).
(c) Based on the result obtain from (b), find the mean and variance of Z.
(d) Find the correlation coefficient between X and Z².
(e) Find the probability of Z ≥ 10.