Let be a random variable. If the expected value exists and is finite for all real numbers belonging to a closed interval , with , then we say that possesses a moment generating function and the functionis called the moment generating function of
DefinitionLet X be a random variable for an experiment taking values in a subset S of R. The moment generating function of X is the function MX defined by
MX(t) = E[exp(tX)] for t in R
PropertiesIn the following exercises, assume that the moment generating functions are finite in an interval about 0.3. Show that for any nonnegative integer n,
MX(n)(0) = E(Xn)Thus, the derivatives of the moment generating function at 0 determine the moments of the variable (hence the name).
4. If a and b are constants, show that
MaX + b(t) = exp(bt) MX(at)5. Suppose that X and Y are independent. Show that
MX + Y(t) = MX(t) MY(t)For more, click here.