Moments generating function

Definition

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

Definition

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 M_X defined by

M_X(t) = E[exp(tX)] for t in R

Properties

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,

M_X⁽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

M_{aX + b}(t) = exp(bt) M_X(at)

 5. Suppose that X and Y are independent. Show that

M_{X + Y}(t) = M_X(t) M_Y(t)

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