What does it mean to write X=Y+W where W is gaussian with mean 0.

Mathematics Asked by zhongyuan chen on January 3, 2022

Assuming X Y are two random variables. When we write X=Y+W with W~Gaussian(with 0 mean), Do we mean the random variable X-Y ~ Gaussian(with 0 mean)? Or does it mean X|y ~ Gaussian(with mean y)? I think the two are clearly not equivalent since the second case seems to imply the first but the first doesn’t imply the second.

One Answer

Probably what was intended is that $Y$ and $W$ are independent of each other and $W$ is Gaussian with expectation $0.$ People often omit to mention such an assumption of independence.

Necessarily this would imply that $X-Y$ is Gaussian with mean $0,$ but people don't usually express the situation that way when they intend the hypothesis of independence of $W$ and $Y$ to be tacitly understood.

The thing that you express by saying $Xmid y simtext{some distribution}$ is something I would express either by saying $Xmid (Y=y) simtext{some distribution depending on $y$}$ (thus $Y$ is a random variable and $y$ is not) or by saying $Xmid Y sim text{some distribution depending on $Y$}$ (so that things like $operatorname E(Xmid Y)$ or $operatorname{var}(Xmid Y)$ would themselves be random variables that are functions of $Y$).

Notice that if you say that $Xmid (Y=y) sim operatorname N(y,sigma^2),$ that will entail that $X-Y$ is independent of $Y,$ as follows: begin{align} Xmid (Y=y) sim operatorname N(y,sigma^2) \[8pt] X-Y mid (Y=y) sim operatorname N(0,sigma^2) end{align} and the expression $text{“}operatorname N(0,sigma^2)text{''}$ has no $text{“}y text{''}$ in it.

Answered by Michael Hardy on January 3, 2022

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