Let $X_{1}, X_{2}$ be independent random variables each from a discrete probability mass function $P_{X}(x) = \begin{cases} 1/3 & \text{if } x = 0 \\ 2/3 & \text{if } x = 1 \end{cases}, i = 1, 2$. Then the moment generating function of $Y = X_{1} \cdot X_{2}$ is
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For binary variables $\{0, 1\}$, the product $X_1 X_2$ behaves like an AND gate in logic. It is only 1 when both inputs are 1. Note that Option (4) is the MGF of the sum $X_1 + X_2$, which is a common trap in these types of questions.