 Solution

Put ${\alpha }_1=\alpha$, ${\alpha }_2=1-\alpha$, and introduce a categorical class label for each observation, encoded by $Z_{i1},Z_{i2}\in \{0,1\}$ with $Z_{i1}+Z_{i2}=1$. Its prior class probability is ${\alpha }_j$, and conditional on class $j$ the observation probability density function is $f_j(x_i)$. Thus its joint observation-label probability density function isIndependence across observations yields the finite Gaussian mixture with a common variance complete-data likelihood functionSumming over the two possible labels at each observation recovers ${\prod }_i[\alpha f_1(x_i)+(1-\alpha )f_2(x_i)]$, the observed-data likelihood function. The product requested in the PDF is therefore a joint complete-data likelihood function. Strictly conditioning on $Z$ would remove the factors ${\alpha }_j$; the PDF's conditional notation for this joint product is imprecise.