Define the moment generating function (Subscript[M, X])(t):

mx[t_]:=(27/40)(((1-(((Exp[t]/3))^4))/(1-Exp[t]/3)))

Differentiate the moment generating function:

mx'[t]

Differentiate and evaluate at t=0. This gives E(X).

mx'[0]

Differentiate twice and evaluate at t=0. This gives E((X^2)).

mx''[0]

Compute the variance of X using the shortcut formula:

3/4-(9/20)^2