Alternative Equation Formats

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

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\begin{align*} \mr{Var}[\bY ] =& \, \, \mr{Cov}[\bY ,\bY ] = \left[\mr{Cov}[Y_ i,Y_ j]\right] \\ =& \, \, \mr{E}\left[ \left(\bY -\mr{E}[\bY ]\right) \left(\bY -\mr{E}[\bY ]\right)^\prime \right] = \mr{E}\left[\mb{YY}'\right] - \mr{E}[\bY ]\mr{E}[\bY ]' \\[0.1in] =& \left[\begin{array}{lllll} \mr{Cov}[Y_1,Y_1] & \mr{Cov}[Y_1,Y_2] & \mr{Cov}[Y_1,Y_3] & \cdots & \mr{Cov}[Y_1,Y_ n] \cr \mr{Cov}[Y_2,Y_1] & \mr{Cov}[Y_2,Y_2] & \mr{Cov}[Y_2,Y_3] & \cdots & \mr{Cov}[Y_2,Y_ n] \cr \mr{Cov}[Y_3,Y_1] & \mr{Cov}[Y_3,Y_2] & \mr{Cov}[Y_3,Y_3] & \cdots & \mr{Cov}[Y_3,Y_ n] \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr \mr{Cov}[Y_ n,Y_1] & \mr{Cov}[Y_ n,Y_2] & \mr{Cov}[Y_ n,Y_3] & \cdots & \mr{Cov}[Y_ n,Y_ n] \cr \end{array}\right] \\[0.15in] =& \left[\begin{array}{lllll} \mr{Var}[Y_1] & \mr{Cov}[Y_1,Y_2] & \mr{Cov}[Y_1,Y_3] & \cdots & \mr{Cov}[Y_1,Y_ n] \cr \mr{Cov}[Y_2,Y_1] & \mr{Var}[Y_2] & \mr{Cov}[Y_2,Y_3] & \cdots & \mr{Cov}[Y_2,Y_ n] \cr \mr{Cov}[Y_3,Y_1] & \mr{Cov}[Y_3,Y_2] & \mr{Var}[Y_3] & \cdots & \mr{Cov}[Y_3,Y_ n] \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr \mr{Cov}[Y_ n,Y_1] & \mr{Cov}[Y_ n,Y_2] & \mr{Cov}[Y_ n,Y_3] & \cdots & \mr{Var}[Y_ n] \cr \end{array}\right] \\ \end{align*}