Exercise 2.3 - Variance of a sum

Answers

Calculate this straightforwardly:

\begin{align} var [X + Y] = & 𝔼 [{(X + Y)}^{2}] - 𝔼^{2} [X + Y] \\ = & 𝔼 [X^{2}] - 𝔼^{2} [X] + 𝔼 [Y^{2}] - 𝔼^{2} [Y] + 2 𝔼 [𝑋𝑌] - 2 𝔼 [X] [Y] \\ = & var [X] + var [Y] + 2 cov [X, Y] . \end{align}

Using the definition of operators var, cov and the linearity of expectation should yield this result easily.

solour_lfq

2021-03-24 13:42