Going over my Lecture Notes my Lecturer as Started with Show that a Gaussian Distribution Corresponds to a CTS random variable. Then she has i) Taken the f(x) = [p.d.f] and shown a) f(x) >= 0 for all x member of real numbers. b) Integral over all real numbers = 1 ii) Found the M.G.F then taken the first two derivatives of MGF and calculated variance. iii) Taken two independent Gaussians and taken a linear combination i.e. aX+bY and found a new mean and variance. My Problems are. 1) How does this shove the initial problem? (this is my only stats module and is this ticking off a definition?) 2) Why Calculate the Variance from the M.G.F 3) What does finding the new mean and variance achieve in case iii) This is a bit of a complicated question any help would be really appreciated. Thanks.