x and y are two vectors of dimension k. Assume that the components of x and y are independent random variables with mean 0 and variance 1. What would be the mean and variance of their dot product, x · y ?
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$\begingroup$ what are the covariances among the elements of $x$ and similarly among the elements of $y$? $\endgroup$– gunesCommented Jul 26, 2020 at 11:37
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$\begingroup$ @gunes The covariances are 0 based on the assumption that all elements are independent random variables. $\endgroup$– MaggieCommented Jul 26, 2020 at 12:52
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$\begingroup$ Are x and y also independent? $\endgroup$– kurtosisCommented Jul 26, 2020 at 16:00
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1 Answer
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If everything is mutually independent, assuming $\sigma^2$ is the common variance (in your case it's $1$), and RVs have zero mean RVs as stated in the OP:
$$\operatorname{var}\left(\sum x_iy_i\right)=k\operatorname{var}(x_1y_1)=kE[x_1^2]E[y_1^2]=k\sigma^4$$
$$E\left[\sum x_iy_i\right]=kE[x_1y_1]=0$$
