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The joint pmf/pdf can be written as a product because the sample observations are assumed to be independent. Independence allows the joint distribution to factor into the product of the corresponding univariate pmf/pdf terms.

Conditions: The observations $x_1, \dots, x_n$ are independent.; The joint distribution is written as $f(x_1, \dots, x_n \mid \theta)$.; Each observation shares the same marginal pmf/pdf $f(x_i \mid \theta)$.

The joint pmf/pdf can be written as a product because the sample observations are assumed to be independent. Independence allows the joint distribution to factor into the product of the corresponding univariate pmf/pdf terms.

Conditions: The observations $x_1, \dots, x_n$ are independent.; The joint distribution is written as $f(x_1, \dots, x_n \mid \theta)$.; Each observation shares the same marginal pmf/pdf $f(x_i \mid \theta)$.