A random variable has the following moment generating functi

A random variable has the following moment generating function Mx(t)= 1/7*(e^t)+2/7*(e^2t)+4/7*(e^3t) find f(x) the PDFs of X, E(X), Var(X)

Mx(t)= E(e^(tx))

therefore from given Mx(t) , x takes values 1,2,3 b/c of e^t,e^3t, e^2t

and pmf at these x = 1/7 , 2/7 , 4/ 7 respectively

therefore pmf(x) = 1/7 x=1

2/7 x=2

4/7 x=3

0 otherwise

E(x)= d/dt(M(t) at t=0

= 1/7 + 2*2/7 + 3*4/7 = 17/7

E(x²)= d²/dt²(M(t) at t=0 = 1/7 + 8/7 + 36/7 = 45/7

V(x)= 45/7 - E(x)^2 = 45/7 - 289/49 = 26/49