 Solution

Write the countable supports as $D_i$ and the marginal masses as $p_i(x_i)$. Independence gives joint mass ${\prod }_i{p}_i(x_i)$. Assuming finite expected values, absolute integrability is justified by Tonelli theorem:Consequently Fubini's theorem permits factorization of the signed sum, proving the expectation of a product of independent random variables:

The second claim holds as a general assertion with the additional hypothesis that the positive variables are integer-valued. For a nonnegative integer-valued $Z$, the pointwise identity $Z={\sum }_{m\ge 0}{1}_{\{Z>m\}}$ and Tonelli theorem give the tail-sum formulaIf each $X_i$ is integer-valued, their product is too. Apply this identity to each factor and to the product, then use independence:Under the literal broader meaning of a discrete random variable, the second claim is false. Two independent constants $X_1=X_2=3/2$ are positive and discrete, but the left side is $2\cdot 2=4$ and the right side is $3$. More generally the integer-indexed tail sum of a positive real-valued variable is $\mathbb{E}\lceil Z\rceil$, not $\mathbb{E}Z$. This is the integer-valued hypothesis in the tail-sum formula.