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? $\endgroup$– gunesCommented Aug 22, 2020 at 16:35
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$\begingroup$ $A$ 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. $\endgroup$– RobCommented Aug 22, 2020 at 16:38
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1 Answer
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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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1$\begingroup$ There is a much easier method requiring almost no calculation. See the follow-up question at stats.stackexchange.com/questions/484222. $\endgroup$– whuber ♦Commented Aug 22, 2020 at 22:00