Properties of Moment-Generating Functions

Two computational properties of moment-generating functions: how the MGF transforms under a linear change of variable, and how the MGF of a sum of independent random variables factors as a product. These let us compute MGFs of complicated combinations from the MGFs of the building blocks.

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Tutorial

MGF of a Linear Transformation

The moment-generating function behaves predictably under a linear change of variable. If $X$ is a random variable with MGF $M_X(t) = E[e^{tX}]$ and $Y = aX + b$ for constants $a,b$ , then

M_{aX+b}(t) = e^{bt} \cdot M_X(at).

This follows directly from the definition:

M_Y(t) = E\!\left[e^{t(aX+b)}\right] = E\!\left[e^{bt} \cdot e^{(at)X}\right] = e^{bt} \cdot E\!\left[e^{(at)X}\right] = e^{bt} M_X(at).

Notice two things: the constant $b$ produces an external factor $e^{bt}$ , and the scalar $a$ rescales the argument of $M_X$ from $t$ to $at$ .

For example, suppose $X$ has MGF $M_X(t) = \dfrac{1}{1-t}$ valid for $t < 1$ . For $Y = 2X + 3$ we have

M_Y(t) = e^{3t} \cdot M_X(2t) = \dfrac{e^{3t}}{1 - 2t}, \qquad t < \tfrac{1}{2}.

The domain changes too: we need $2t < 1$ , so the new MGF is valid for $t < 1/2$ .