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Homework 4.1 Independent and uncorrelated random variables. Consider a two independent zero mean Gaussian random variable $X_{1}$ and $X_{2}$ with variance $\sigma^{2}$. Let $Y_{1} \stackrel{\text { def }}{=} X_{1}+X_{2}$ and $Y_{2} \stackrel{\text { def }}{=} X_{1}+a X_{2}$. What are the values of $a$ for which $Y_{1}$ and $Y_{2}$ are independent?

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