A random variable Y has the following probability distributi

A random variable Y has the following probability distribution of y=1,2,3 and p(y)=0.1, 0.5, 0.4.

a) Find the moment generating funtion of mgf for Y.

b) Find E[Y²] using the mgf, m(t)? Explain.

Solution

(a)\\(M_Y(t) = E[e^{tY}] = 0.1 e^{t} + 0.5 e^{2t} + 0.4 e^{3t}\\)  

(b)   \\(E[Y] = M\'(0)\\)  

  \\(M\'(t) = 0.1 e^t + 1 \\cdot e^{2t} + 1.2 e^{3t}\\)  

So

  \\(E[Y] = M\'(0) = 0.1 + 1 + 1.2 = 2.3\\)