Why Is the Expected Value a Linear Operator in Quantum Mechanics?

[KastorPhys]

In a Quantum Mechanics Class, my tutor had shown me that
the variance Δa²

Δa² = < ( a - < a > )² >
= < a² - 2 a < a > + < a >² >
= < a² > - 2 < a >< a > + < a >²
= < a² > - 2 < a >² + < a >²
= < a² > - < a >²

and my questions are

i) why, in step 3,

< x - y + z > will become < x > - < y > + < z > ?

ii)why, also in step 3, the middle term,

< -2 a < a > > will finally become -2 < a >² ?

Is it mean that < a < a > > = < a > < a > = < a >² ?? but why?

Thanks!

 

[Mark44]

In a Quantum Mechanics Class, my tutor had shown me that
the variance Δa²

Δa² = < ( a - < a > )² >
= < a² - 2 a < a > + < a >² >
= < a² > - 2 < a >< a > + < a >²
= < a² > - 2 < a >² + < a >²
= < a² > - < a >²

and my questions are

i) why, in step 3,

< x - y + z > will become < x > - < y > + < z > ?

ii)why, also in step 3, the middle term,

< -2 a < a > > will finally become -2 < a >² ?

Is it mean that < a < a > > = < a > < a > = < a >² ?? but why?

Thanks!

I'm not familiar with your notation. One definition of the variance of a random variable X (from Wikipedia - see https://en.wikipedia.org/wiki/Variance) is:
$Var(X) = E[(X - \mu)^2]$
Here E[X] means the expected value of X.

The right side of the equation above can be expanded to yield
$Var(X) = E[(X - E[X])^2] \\ = E[X^2 - 2XE[X] + (E[X])^2] \\ = E[X^2 - 2E[X] E[X] +(E[X])^2] \\ = E[X^2] - (E[X])^2$

 

[pwsnafu]

In all three questions the answer is the expected value is a linear operator. For an observable $A$ the expected value is defined as
$\langle A \rangle _{\phi} := \langle \phi | \, A \,| \phi \rangle$. Suppose that $A$ and $B$ are observables (mathematically they are operators on some Hilbert space), and let $\alpha$ and $\beta$ be complex numbers. Then
$\langle \alpha A + \beta B \rangle_\phi = \langle \phi | \alpha A + \beta B | \phi \rangle = \langle \phi | \alpha A | \phi \rangle + \langle \phi | \beta B | \phi \rangle =\alpha \langle \phi | A | \phi \rangle + \beta \langle \phi | B | \phi \rangle = \alpha \langle A \rangle_\phi + \beta \langle A \rangle_\phi$