Alternative Equation Formats

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MathML

\begin{aligned} C o v [Y, U] = & [C o v [Y_{i}, U_{j}]] \\ = & E [(Y - E [Y]) {(U - E [U])}^{'}] = E [{Y U}^{'}] - E [Y] E [U]^{'} \\ = & [\begin{array}{lllll} C o v [Y_{1}, U_{1}] & C o v [Y_{1}, U_{2}] & C o v [Y_{1}, U_{3}] & \dots & C o v [Y_{1}, U_{k}] \\ C o v [Y_{2}, U_{1}] & C o v [Y_{2}, U_{2}] & C o v [Y_{2}, U_{3}] & \dots & C o v [Y_{2}, U_{k}] \\ C o v [Y_{3}, U_{1}] & C o v [Y_{3}, U_{2}] & C o v [Y_{3}, U_{3}] & \dots & C o v [Y_{3}, U_{k}] \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C o v [Y_{n}, U_{1}] & C o v [Y_{n}, U_{2}] & C o v [Y_{n}, U_{3}] & \dots & C o v [Y_{n}, U_{k}] \end{array}] \end{aligned}

MathML rendered with MathJax

\begin{aligned} C o v [Y, U] = & [C o v [Y_{i}, U_{j}]] \\ = & E [(Y - E [Y]) {(U - E [U])}^{'}] = E [{Y U}^{'}] - E [Y] E [U]^{'} \\ = & [\begin{array}{lllll} C o v [Y_{1}, U_{1}] & C o v [Y_{1}, U_{2}] & C o v [Y_{1}, U_{3}] & \dots & C o v [Y_{1}, U_{k}] \\ C o v [Y_{2}, U_{1}] & C o v [Y_{2}, U_{2}] & C o v [Y_{2}, U_{3}] & \dots & C o v [Y_{2}, U_{k}] \\ C o v [Y_{3}, U_{1}] & C o v [Y_{3}, U_{2}] & C o v [Y_{3}, U_{3}] & \dots & C o v [Y_{3}, U_{k}] \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C o v [Y_{n}, U_{1}] & C o v [Y_{n}, U_{2}] & C o v [Y_{n}, U_{3}] & \dots & C o v [Y_{n}, U_{k}] \end{array}] \end{aligned}

Math as LaTeX Source

\begin{align*} \mr{Cov}[\bY ,\bU ] =& \left[\mr{Cov}[Y_ i,U_ j]\right] \\ =& \, \, \mr{E}\left[ \left(\bY -\mr{E}[\bY ]\right) \left(\bU -\mr{E}[\bU ]\right)^\prime \right] = \mr{E}\left[\mb{YU}'\right] - \mr{E}[\bY ]\mr{E}[\bU ]' \\[0.1in] =& \left[\begin{array}{lllll} \mr{Cov}[Y_1,U_1] & \mr{Cov}[Y_1,U_2] & \mr{Cov}[Y_1,U_3] & \cdots & \mr{Cov}[Y_1,U_ k] \cr \mr{Cov}[Y_2,U_1] & \mr{Cov}[Y_2,U_2] & \mr{Cov}[Y_2,U_3] & \cdots & \mr{Cov}[Y_2,U_ k] \cr \mr{Cov}[Y_3,U_1] & \mr{Cov}[Y_3,U_2] & \mr{Cov}[Y_3,U_3] & \cdots & \mr{Cov}[Y_3,U_ k] \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr \mr{Cov}[Y_ n,U_1] & \mr{Cov}[Y_ n,U_2] & \mr{Cov}[Y_ n,U_3] & \cdots & \mr{Cov}[Y_ n,U_ k] \cr \end{array}\right] \end{align*}