= Solution
Write the countable supports as $D_i$ and the marginal masses as $p_i(x_i)$. <Independence> gives joint mass $\prod_ip_i(x_i)$. Assuming finite <expected values>, absolute integrability is justified by <Tonelli theorem>:
$$
\mathbb E\left[\prod_i|X_i|\right]=\sum_{x_1\in D_1,\ldots,x_n\in D_n}\prod_i|x_i|p_i(x_i)=\prod_i\mathbb E|X_i|<\infty.
$$
Consequently <Fubini's theorem> permits factorization of the signed sum, proving the <expectation of a product of independent random variables>:
$$
\boxed{\mathbb E\left[\prod_iX_i\right]=\prod_i\mathbb E[X_i].}
$$
The second claim holds as a general assertion with the additional hypothesis that \b[the positive variables are integer-valued]. For a nonnegative integer-valued $Z$, the pointwise identity $Z=\sum_{m\ge0}\mathbf1_{\{Z>m\}}$ and <Tonelli theorem> give the <tail-sum formula>
$$
\mathbb E[Z]=\sum_{m=0}^\infty\mathbb P(Z>m).
$$
If each $X_i$ is integer-valued, their product is too. Apply this identity to each factor and to the product, then use <independence>:
$$
\boxed{\prod_i\sum_{m\ge0}\mathbb P(X_i>m)=\prod_i\mathbb E[X_i]=\mathbb E\left[\prod_iX_i\right]=\sum_{m\ge0}\mathbb P\left(\prod_iX_i>m\right).}
$$
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 $2\cdot2=4$ 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>.
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