# Variance of the product of a unit vector with a scalar

How can we calculate

$$Var(AX)$$

if $$A$$ is a unit vector with a uniform random phase between $$[0,2\pi]$$ and $$X$$ is a scalar random variable with a uniform distribution on $$[-1,1]$$. $$A$$ and $$X$$ are independent. Thank you.

• What are the dimensions of $A$ and $X$, and how did you start solving it? – gunes Aug 22 '20 at 16:35
• $A$ is a $2d$ unit vector and $X$ is a scalar. Since i don't know anything else about $A$ i wonder if those two points( its unity and its phase) can simplify the problem. – Rob Aug 22 '20 at 16:38

Since $$A=[\cos(\theta)\ \sin(\theta)]^T$$, where $$\theta\sim U[0,2\pi], \operatorname{var}(AX)$$ is a $$2 \times 2$$ covariance matrix. It's first (upper-left) entry is the variance of $$\cos(\theta)X$$:
$$\operatorname{var}(\cos(\theta)X)=\mathbb E[\cos^2\theta]\mathbb E[X^2]-\mathbb E[\cos\theta]^2\mathbb E[X]^2=E[\cos^2\theta]\mathbb E[X^2]$$
I think you can find $$E[\cos^2\theta]$$ and $$\mathbb E[X^2]$$ without much trouble using their PDFs. Other entries' calculations will be similar as well.