In each example below find the probability generating function (p.g.f.) or moment generating function (m.g.f.) of the random variable \(X\): (show your work!) ( When both are asked for, recall the relationship between the m.g.f. and p.g.f. You only need to do the calculation for one to find both.)
- \(X\) is normal mean \(\mu\) and variance \(\sigma^2\). Find m.g.f.
- \(X\) is uniform on \( (0,a)\). Find m.g.f.
- \(X\) is Bernoulli with parameter \(p\). Find p.g.f. and m.g.f.
- \(X\) is exponential with parameter \(\lambda\). Find m.g.f.
- \(X\) is geometric with parameter \(p\). Find p.g.f. and m.g.f.
- \(X=a+bY\) where \(Y\) has probability generating function \(G(s)\). Find m.g.f.
