For independent random variables, the moment generating function of a sum is the product of the individual MGFs: $M_{X+Y}(\theta) = M_X(\theta) \cdot M_Y(\theta)$. This is the exponential-transform analogue of the [PGF of a Sum](/theorems/1130) and is proved by the same mechanism: writing $e^{\theta(X+Y)} = e^{\theta X} \cdot e^{\theta Y}$ and applying the [Expectation of a Product](/theorems/1120) to the independent factors $e^{\theta X}$ and $e^{\theta Y}$. This multiplicativity is the primary tool for determining the distribution of sums of independent random variables when the MGF is available. Combined with the [MGF Determines the Distribution](/theorems/1142) theorem, it shows for instance that a sum of independent normals $N(\mu_i, \sigma_i^2)$ is again normal: the product $\prod e^{\mu_i \theta + \sigma_i^2 \theta^2/2} = e^{(\sum \mu_i)\theta + (\sum \sigma_i^2)\theta^2/2}$ is recognised as the MGF of $N(\sum \mu_i, \sum \sigma_i^2)$. Similarly, the Poisson and gamma addition rules follow by multiplying MGFs and identifying the product. This approach extends by induction to any finite sum of independent variables and forms the basis of the [Continuity Theorem](/theorems/1145), which uses MGF convergence to establish the Central Limit Theorem.