How to find the variance of a "function"

Question

Here's the problem:

I have $n$ i.i.d. rvs. $X_1,\ldots,X_n$. All coming from $ N(\theta, 1)$.

$$\bar{X} = \frac{1}{n} \sum_1^n X_i$$

$c$ is a constant.

How would I find:

$$\operatorname{var}(c\bar{X} - \theta)$$

Answer 1

$\newcommand{\var}{\operatorname{var}}$ $$ \var(c\bar X-\theta) = \var(c\bar X) = c^2 \var(\bar X) $$ $$ = c^2 \var\left(\frac{X_1+\cdots+X_n}{n}\right) = \frac{c^2}{n^2}\var(X_1+\cdots+X_n) $$ $$ =\frac{c^2}{n^2}\left(\var(X_1)+\cdots+\var(X_n)\right) $$ $$ =\frac{c^2}{n^2}\cdot n \var(X_1) = \frac{c^2}{n} \var(X_1) = \frac{c^2}{n}. $$