EQ7 Expectation values

1.

Suppose that g₁ and X while $\alpha$ is a constant. Deduce that

$$\langle g_1(x) + g_2(x) \rangle = \langle g_1(x) \rangle +\langle g_2(x) \rangle$$

and

$$\langle \alpha g(x) \rangle = \alpha \langle g(x) \rangle$$

2.

Show that

$$\langle \Delta x^2 \rangle = \langle x^2 \rangle -\langle x ... ...{\rm where}} \hspace*{1cm} \Delta x \equiv x-\langle x \rangle$$

3.

Show that if X and Y are independent random variables

$$\langle xy \rangle =\langle x \rangle \langle y \rangle$$