calculate moment generating function for poisson9 and binomi

calculate moment generating function for poisson(9) and binomial (3,0.3) and geometric (0.9). Binomial (n,p): f(x) = (nx ) p^x (1 - p)^n-x E(X) = np Var(X) = np(1 - p) Mx(t) = (pe^t + (1-p))^n Geometric(p): F(x) = p (1 -p)^x-1 E (X) = 1/p Var (X) = 1-p/p^2 Mx (t) = pe^t/1- (1-p)e^t Poisson (Lambda) F (x) = lambda ^x/x! e^- lambda E (X) = Lambda Var (X) = Lambda Mx (t) = exp ( lambda (e^t -1))

Solution

mean = 9

M(t) = exp( mean * (e^t -1))

so, M(t)= exp [ 9*(e^t -1)]

2.

n= 3

p = 0.3

M(t) = [ p*e^t + (1-p) ]^n

so, M(t) = [ 0.3e^t + 0.7 ]^3

3.

p=0.9

so, M(t) = pe^t / [1 - (1-p) e^t ]

M(t) = 0.9e^t / [ 1- 0.1 e^t]