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$X$ and $Y$ are random variables uniformly distributed on $[a,b]$. Could the random variable $Z=X+Y$ be uniformly distributed if $X$ and $Y$ are dependent and correlation between them doesn't equal $1$ or $-1$?

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    $1/12$ of the square of the range of $Z$ is the variance of $Z$ if $Z$ is uniformly distibuted. That means the range of $Z$ would be less than $2(b-a)$. The variance of $Z$ is also $(b-a)^2(1+\rho)$ where $\rho$ is the correlation. Shortening the range of $Z$ necessarily means chopping triangles off corners of the square $[a,b]^2$.2012-12-27

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