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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.

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  • $\begingroup$ What are the dimensions of $A$ and $X$, and how did you start solving it? $\endgroup$ – gunes Aug 22 '20 at 16:35
  • $\begingroup$ $A$ 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. $\endgroup$ – Rob Aug 22 '20 at 16:38
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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.

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