Let X be a r.v has a p.f defined as: P(x)={(3(1/4)^x , X=1,2,3,… 0 otherwise)┤ Find the m.g.f, and then deduce the S.D

∵M_x (t)=E(e^tx )=∑_x e^tx  P(x)
∴M_x (t)=∑_(x=1)^∞ 〖3(1/4)^x 〗 e^tx=3∑_(x=1)^∞ (e^t/4)^x
=3[e^t/4+(e^t/4)^2+⋯]
=3 e^t/4 [1+e^t/4+(e^t/4)^2+⋯]
3 e^t/4 [1/(1-e^t/4)]=〖3e〗^t/4 [4/(4-e^t )]=3e^t (4-e^t )^(-1)
∴M_x (t)=3e^t (4-e^t )^(-1) This is the required m.g.f
To evaluate the S.D:
μ_1=μ=├ (dM_x (t))/dt┤|_(t=0)
μ_1={3e^t (4-e^t )^(-1)+3e^t [-(4-e^t )^(-2) (-e^t )]}_(t=0)
=[3e^t (4-e^t )^(-1)+e^t (4-e^t )^(-2) ]_(t=0)=1+3/9=  
∵μ_2=├ (d^2 M_x (t))/dt┤|_(t=0)
∴μ_2={3e^t [(4-e^t )^(-1)+(4-e^t )^(-2) e^t ]+3e^t [e^t (4-e^t )^(-2)+2e^t (4-e^t )^(-3)+e^t (4-e^t )^(-2) ]}_(t=0)
∴μ_2=3(1/3+1/9)+3(2/9+2/27)=4/3+8/9=20/9
And the variance becomes:
σ_x^2=μ_2-μ_1^2=20/9-16/9=4/9
Thus the S.D is:  σ_x=√(4/9)=2/3