Question

Let X be a random variable such that -1, with probability q 1 with probability 1-q,

a) Find the moment generating function (mgf) of X.

b) Using part a), that is the mgf of X, find the expected value (E[X]) and (V ar[X]).

Answer 1

a)

as we know that mgf=M_x(t)=$\sum$ e^txP(x)

M_x(t) =e^-t*q+e^t*(1-q)

b)

first derivation of mgf :M'_x(t)=(d/dt)*(e^-t*p+e^t*(1-p))=-qe^-t+(1-q)e^t

hence E(X)=M'_x(0)=-qe^-0+(1-q)e⁰ =-q+1-q =1-2q

second derivation of mgf :M''_x(t)=(d/dt)*(-qe^-t+(1-q)e^t )=qe^-t+(1-q)e^t

E(X²)=:M''_x(0)=qe^-0+(1-q)e⁰ =q+1-q =1

therefore Var(X)=E(X²)-(E(X))² =1-(1-2q)² =4q-4q² =4q(1-q)