EM for an independent missing normal coordinate

For independent normal coordinates, an unobserved first coordinate has conditional law equal to its current marginal normal distribution, even when the second coordinate of that observation is known. Thus its E-step first moment is ${\mu }_1^{(t)}$ and second moment is $({\mu }_1^{(t)}{)}^2+v_1^{(t)}$. With three observed first coordinates and one missing value, the mean update is $({\sum }_{i=1}^3{x}_{i1}+{\mu }_1^{(t)})/4$ and the variance update is $[{\sum }_{i=1}^3(x_{i1}-{\mu }_1^{(t+1)}{)}^2+v_1^{(t)}+({\mu }_1^{(t)}-{\mu }_1^{(t+1)}{)}^2]/4$. The conditional variance term is essential: filling in only the conditional mean is not the expectation-maximization algorithm.