Moments generating function


Let X be a random variable. If the expected value [eq1] exists and is finite for all real numbers $t$ belonging to a closed interval [eq2], with $h>0$, then we say that X possesses a moment generating function and the function[eq3]is called the moment generating function of X


Let 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


In 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)
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