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Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 +bX + c) = a\mathrm{M}(X^2) +b\mathrm{M}(X) + c$

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 +bX + c) = a\mathrm{M}(X^2) +b\mathrm{M}(X) + c$

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 +bX + c) = a\mathrm{M}(X^2) +b\mathrm{M}(X) + c$c)$

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 +bX + c)$$\mathrm{M}$

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}$$\mathrm{M}(aX^2 +bX + c) $

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 +bX + c) $\mathrm{M}(a*X^2) $

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(a*X^2) $\mathrm{M}(aX^2 + b X) $

and so on for all polynomial ?

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

$\mathrm{M}(aX^2 + b X) $

and so on other result for all polynomial ?variances and covariances

Is there a way to define as a new operator the mathematical expectation.

Say we call it $\mathrm{M}$ as in the old Mir (russian books).

Wa We should have

$\mathrm{M}(kX) = k\mathrm{M}(X)$

$\mathrm{M}(X \pm Y) = \mathrm{M}(X)\pm \mathrm{M}(Y)$

and as a résult

$\mathrm{M}(X -\mathrm{M}X)^2 =\mathrm{M}(X^2)- \mathrm{M}(X)^2$

and other result for variances and covariances