Question 7.10: Consider n i.i.d. random variables Xi with E(Xi ) = μ and Va......

If we evaluate the expectation with respect to Y , then both μ and σ can be considered to be constants. We can therefore write

E(Y) = E \left(\frac{X−\mu}{\sigma }\right)= \frac{1}{\sigma }\left(E\left(X\right) − \mu\right).

Since E(X) = μ, it follows that E(Y ) = 0. The variance is

Var(Y ) = Var \left(\frac{X−\mu}{\sigma }\right).

Applying Var(a + bX) = b²Var(X) to this equation yields a = μ, b = \frac{1}{\sigma } and therefore

Var(Y ) = \frac{1}{\sigma^{2} } Var(X) = \frac{\sigma^{2}}{\sigma^{2} }=1.